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# Theoretical properties of angular halfspace depth

Stanislav Nagy1 and  Petra Laketa1<EMAIL_ADDRESS>1 Faculty of
Mathematics and Physics, Charles University, Prague, Czech Republic

###### Abstract.

The angular halfspace depth (ahD) is a natural modification of the celebrated
halfspace (or Tukey) depth to the setup of directional data. It allows us to
define elements of nonparametric inference, such as the median, the inter-
quantile regions, or the rank statistics, for datasets supported in the unit
sphere. Despite being introduced in 1987, ahD has never received ample
recognition in the literature, mainly due to the lack of efficient algorithms
for its computation. With the recent progress on the computational front, ahD
however exhibits the potential for developing viable nonparametric statistics
techniques for directional datasets. In this paper, we thoroughly treat the
theoretical properties of ahD. We show that similarly to the classical
halfspace depth for multivariate data, also ahD satisfies many desirable
properties of a statistical depth function. Further, we derive uniform
continuity/consistency results for the associated set of directional medians,
and the central regions of ahD, the latter representing a depth-based analogue
of the quantiles for directional data.

##  1\. Nonparametrics of directional data and angular halfspace depth

Directional data analysis concerns the statistics of datasets bound to lie on
the unit sphere
$\mathbb{S}^{d-1}=\left\\{x\in\mathbb{R}^{d}\colon\left\|x\right\|=1\right\\}$.
Despite sharing similarities with multivariate statistical methods, the
particular geometry of the sphere makes the analysis of directional data
challenging. The sphere $\mathbb{S}^{d-1}$ is a symmetric compact manifold,
which makes many statistical methods from $\mathbb{R}^{d}$ fruitless or
suboptimal when applied in $\mathbb{S}^{d-1}$ directly (Watson,, 1983, Mardia
and Jupp,, 2000, Ley and Verdebout,, 2017).

We consider the nonparametric analysis for directional data and the concept of
depth functions, a statistical tool that introduces elements of nonparametrics
to multivariate or non-Euclidean spaces. In the past decades, depths have
garnered great success in multivariate analysis (Donoho and Gasko,, 1992, Liu
et al.,, 1999, Zuo and Serfling,, 2000, Mosler,, 2013, Chernozhukov et al.,,
2017, Mosler and Mozharovskyi,, 2022). A prime example of a depth in the
Euclidean space $\mathbb{R}^{d}$ is the _halfspace depth_ (hD, also called
_Tukey depth_ , or _location depth_) that is defined for $x\in\mathbb{R}^{d}$
and a Borel probability measure $P$ on $\mathbb{R}^{d}$ by

(1) $hD(x;P)=\inf\left\\{P\left(H_{x,u}\right)\colon
u\in\mathbb{S}^{d-1}\right\\},$

where $H_{x,u}=\left\\{y\in\mathbb{R}^{d}\colon\left\langle
y,u\right\rangle\geq\left\langle x,u\right\rangle\right\\}$ is the closed
halfspace whose boundary passes through $x$ with inner unit normal $u$. In
words, hD evaluates the smallest $P$-mass of a halfspace that contains $x$. As
such, it allows to devise a $P$-dependent ordering of the points from
$\mathbb{R}^{d}$ — a point $x$ is said to be located deeper than $y$ inside
the mass of $P$ if $hD(x;P)>hD(y;P)$. The deepest point in $\mathbb{R}^{d}$,
defined as (any) point that maximizes the depth function $x\mapsto hD(x;P)$
over $\mathbb{R}^{d}$, is a natural analogue of the median for
$\mathbb{R}^{d}$-valued measures and is called a _halfspace median_ of $P$. A
counterpart of quantiles (or, more precisely, inter-quantile regions) in
$\mathbb{R}^{d}$ are the _central regions_ of $P$, given as the upper level
sets of hD

(2) $hD_{\alpha}(P)=\left\\{x\in\mathbb{R}^{d}\colon
hD(x;P)\geq\alpha\right\\}\quad\mbox{for }\alpha\geq 0.$

These sets are known to be nested, convex, and compact for $\alpha>0$; their
shapes capture the geometry of $P$. Both the multivariate medians and central
regions are of great importance in nonparametric analysis of multivariate
data, and have been studied extensively in the literature (Donoho and Gasko,,
1992, Rousseeuw and Ruts,, 1999, Zuo and Serfling,, 2000, Dyckerhoff,, 2017).
The halfspace depth is only one of many depth functions proposed in
multivariate analysis. It is, nevertheless, the classical representative of
multivariate depths.

Several depths suitable specifically for directional data have been proposed
in the literature; we refer to Liu and Singh, (1992), Agostinelli and
Romanazzi, (2013), Ley et al., (2014), Pandolfo et al., (2018), Buttarazzi et
al., (2018), Konen, (2022), and Hallin et al., (2022). In this contribution,
we scrutinize historically the first directional depth function proposed by
Small, (1987, Example 2.3.4). It is a version of hD from (1) suitable for
measures in $\mathbb{S}^{d-1}$. For $x\in\mathbb{S}^{d-1}$ and $P$ a Borel
probability measure on $\mathbb{S}^{d-1}$, the _angular halfspace depth_ (ahD,
also known as _angular Tukey depth_) of $x$ with respect to (w.r.t.) $P$ is
defined as

(3) $ahD\left(x;P\right)=\inf\left\\{P\left(H_{0,u}\right)\colon
u\in\mathbb{S}^{d-1}\mbox{ and }x\in H_{0,u}\right\\}.$

In contrast to the standard hD in (1), in the definition of ahD one considers
only halfspaces whose boundary passes through the origin $0$ in the ambient
space $\mathbb{R}^{d}$, and searches for a minimum $P$-mass among those that
contain $x$.

The first rigorous studies of ahD were conducted by Small, (1987) and Liu and
Singh, (1992, Section 4). Since then, no systematic investigation of the
theory of ahD has been performed. In fact, it might come as a surprise how
little attention did ahD receive in the literature, especially when compared
with the abundant body of research on the classical hD. One explanation for
this is the presumed high computational cost of the angular depth, coupled
with a lack of efficient algorithms for its computation (Pandolfo et al.,,
2018). The problem of exact and approximate computation of ahD was, however,
recently resolved in Dyckerhoff and Nagy, (2023), which paved the way to
explore the general theory and practice of ahD with its statistical
applications.

This paper comprehensively studies the main theoretical properties of ahD, its
associated median, and its central regions. We contrast ahD with hD and
demonstrate that similarly to the halfspace depth in $\mathbb{R}^{d}$, also
the angular depth satisfies an array of plausible properties required from a
proper depth function. After introducing the notations, we begin in Section 2
by drawing a direct relation between ahD on the sphere $\mathbb{S}^{d-1}$ and
a variant of hD in $\mathbb{R}^{d-1}$. Then, in Section 3, we study ahD w.r.t.
the desirable properties of a directional depth formulated recently in Nagy et
al., (2023). We show that ahD, as the only directional depth function found in
the literature, satisfies all the conditions from Nagy et al., (2023). In the
final Section 4 we provide a list of additional characteristics of ahD, mainly
related to the continuity of its median and the associated central regions.
Those findings are important in statistical practice, as they guarantee the
uniform consistency of the sample depth (3) when computed w.r.t. the empirical
measure of a random sample $X_{1},\dots,X_{n}$ from distribution $P$. The
proofs of all theoretical results are gathered in the Appendix.

### Notations

We write $e_{j}\in\mathbb{S}^{d-1}$ for the $j$-th canonical vector in
$\mathbb{R}^{d}$, $j=1,\dots,d$. That is, $e_{1}=(1,0,\dots,0)^{\mathsf{T}}$
etc. We denote by $\mathcal{H}=\left\\{H_{x,u}\colon x\in\mathbb{R}^{d}\mbox{
and }u\in\mathbb{S}^{d-1}\right\\}$ the collection of all halfspaces in
$\mathbb{R}^{d}$, and by $\mathcal{H}_{0}=\left\\{H_{0,u}\in\mathcal{H}\colon
u\in\mathbb{S}^{d-1}\right\\}$ those halfspaces in $\mathbb{R}^{d}$ whose
boundary passes through the origin. For a set $A\subset\mathbb{R}^{d}$, we
write $A^{\circ}$ for the interior of $A$ and $\partial A$ for the topological
boundary of $A$. Sometimes we use $A^{\circ}$ ($\partial A$) to denote the
relative interior (boundary) of $A$, that is the interior (boundary)
considered in the smallest affine subspace containing $A$. Whether we consider
relative interior (boundary) or not will always be clear from the context. The
complement of $A$ is $A^{\mathsf{c}}$. A set $A\subseteq\mathbb{S}^{d-1}$ is
called _spherical convex_ (see, e.g., Besau and Werner,, 2016) if its radial
extension defined as
$\mathrm{rad}\left(A\right)=\left\\{\lambda\,a\in\mathbb{R}^{d}\colon a\in
A\mbox{ and }\lambda>0\right\\}$ is convex in $\mathbb{R}^{d}$. We denote by
$\mathbb{S}^{d-1}_{+}=\left\\{y\in\mathbb{S}^{d-1}\colon\left\langle
y,e_{d}\right\rangle>0\right\\}$ and
$\mathbb{S}^{d-1}_{-}=\left\\{y\in\mathbb{S}^{d-1}\colon\left\langle
y,e_{d}\right\rangle<0\right\\}$ the (open) “northern” hemisphere with a pole
at $e_{d}$, and the (open) “southern” hemisphere centered at $-e_{d}$,
respectively. The set
$\mathbb{S}^{d-1}_{0}=\left\\{y\in\mathbb{S}^{d-1}\colon\left\langle
y,e_{d}\right\rangle=0\right\\}$ is called the “equator” of the $(d-1)$-sphere
$\mathbb{S}^{d-1}$.

Let $\left(\Omega,\mathcal{A},\mathsf{P}\right)$ be the probability space on
which all random variables are defined. For a topological space $\mathcal{X}$,
$\mathcal{P}\left({\mathcal{X}}\right)$ stands for the collection of all Borel
probability measures on $\mathcal{X}$, and $X\sim
P\in\mathcal{P}\left({\mathcal{X}}\right)$ means that $X$ is a random variable
in $\mathcal{X}$ with distribution $P$. For
$\varphi\colon\mathcal{X}\to\mathcal{Y}$ a map between topological spaces, we
write $P_{\varphi(X)}\in\mathcal{P}\left({\mathcal{Y}}\right)$ for the
distribution of $\varphi(X)$ with $X\sim
P\in\mathcal{P}\left({\mathcal{X}}\right)$. Further,
$\mathcal{M}\left(\mathcal{X}\right)$ represents the collection of all finite
Borel measures on $\mathcal{X}$. Certainly,
$\mathcal{P}\left({\mathcal{X}}\right)\subset\mathcal{M}\left(\mathcal{X}\right)$.
Weak convergence of a sequence of measures
$\left\\{P_{n}\right\\}_{n=1}^{\infty}\subset\mathcal{M}\left(\mathcal{X}\right)$
to $P\in\mathcal{M}\left(\mathcal{X}\right)$ is denoted by
$P_{n}\xrightarrow{w}P$ as $n\to\infty$. Finally, we say that $Q$ is a (finite
Borel) _signed measure_ on a topological space $\mathcal{X}$ if there exist
two finite Borel measures $Q_{+},Q_{-}\in\mathcal{M}\left(\mathcal{X}\right)$
such that (Dudley,, 2002, Theorem 5.6.1)

(4) $Q(B)=Q_{+}(B)-Q_{-}(B)\quad\mbox{for all $B\subseteq\mathcal{X}$ Borel}.$

A signed measure can attain both positive and negative values. We denote by
$\mathcal{S}\left({\mathcal{X}}\right)$ the set of all finite Borel signed
measures on $\mathcal{X}$, and note that
$\mathcal{M}\left(\mathcal{X}\right)\subset\mathcal{S}\left({\mathcal{X}}\right)$.

##  2\. Gnomonic projection and $ahD$

We begin our study by drawing connections of ahD in $\mathbb{S}^{d-1}$ with
the standard (Euclidean) hD in $\mathbb{R}^{d-1}$. They will allow us to use
the abundance of theoretical results available for hD and adapt them to the
setup of directional measures. First, note that the halfspace depth (1) is
well defined not only for probability measures but for any (finite) Borel
measures $P\in\mathcal{M}\left(\mathbb{R}^{d}\right)$. It can be written in
two equivalent forms, either as

(5) $hD\left(x;P\right)=\inf\left\\{P\left(H\right)\colon
H\in\mathcal{H}\mbox{ and }x\in H\right\\},$

or also as

(6) $hD\left(x;P\right)=\inf\left\\{P\left(H\right)\colon
H\in\mathcal{H}\mbox{ and }x\in\partial H\right\\}.$

In what follows, we employ the halfspace depth with signed measures. In case
when $Q\in\mathcal{S}\left({\mathbb{R}^{d}}\right)$ can attain negative
values, the two definitions of hD in (5) and (6) differ; a halfspace $H$
containing $x$ in its interior may have strictly smaller $Q$-mass than its
subset $H^{\prime}$ with $x\in\partial H^{\prime}$. Out of the two
possibilities of defining hD for signed measures, it will be convenient to use
the first one in (5). We say that the _halfspace depth_ of
$x\in\mathbb{R}^{d}$ w.r.t. $Q\in\mathcal{S}\left({\mathbb{R}^{d}}\right)$ is
defined as

(7) $hD\left(x;Q\right)=\inf\left\\{Q\left(H\right)\colon
H\in\mathcal{H}\mbox{ and }x\in H\right\\}.$

We use the same notation for both halfspace depth of measures from
$\mathcal{M}\left(\mathbb{R}^{d}\right)$ and signed measures from
$\mathcal{S}\left({\mathbb{R}^{d}}\right)$; the standard depth (5) is only a
particular case of our more general depth (7).

Our task is to express ahD of $x\in\mathbb{S}^{d-1}$ w.r.t.
$P\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$.111We could equally work with
$P\in\mathcal{M}\left(\mathbb{S}^{d-1}\right)$ without having to restrict to
probability measures. All our results also work for
$P\in\mathcal{M}\left(\mathbb{S}^{d-1}\right)$, with obvious minor
modifications. Throughout this section, we make the following assumption on
$P$ and $x$

(8) $P(\mathbb{S}^{d-1}_{0})=0\quad\mbox{and}\quad x\in\mathbb{S}^{d-1}_{+}.$

This assumption is made without loss of generality due to the rotational
invariance of ahD that will be formally proved as Theorem 2 in Section
3.1.222Indeed, if (8) is not valid for
$Q\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$ and we want to compute
$ahD(z;Q)$ for $z\in\mathbb{S}^{d-1}$, it is always possible to find a
direction $u\in\mathbb{S}^{d-1}$ such that
$Q(\left\\{y\in\mathbb{S}^{d-1}\colon\left\langle
y,u\right\rangle=0\right\\})=0$ and $\left\langle y,z\right\rangle>0$.
Applying (any) orthogonal rotation $O\in\mathbb{R}^{d\times d}$ such that
$O\,e_{d}=u$ to $X\sim Q$, we obtain
$P\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$, with $P$ the measure
corresponding to the random vector $OX$. Then, (8) is true, and
$x=Oz\in\mathbb{S}^{d-1}_{+}$. By Theorem 2 we have $ahD(x;P)=ahD(z;Q)$, while
the conditions (8) are now valid. Denote by
$\mathcal{G}=\left\\{y\in\mathbb{R}^{d}\colon\left\langle
y,e_{d}\right\rangle=1\right\\}$ the hyperplane tangent to $\mathbb{S}^{d-1}$
at the pole $e_{d}\in\mathbb{R}^{d}$, see Figure 1. Consider the mapping

(9)
$\xi\colon\mathbb{S}^{d-1}\setminus\mathbb{S}^{d-1}_{0}\to\mathcal{G}\colon
x\mapsto x/\left\langle x,e_{d}\right\rangle$

that takes points $x\in\mathbb{S}^{d-1}$ not on the equator to the unique
point of $\mathcal{G}$ on the straight line between $x$ and the origin. This
map is visualized for $d=2$ and $d=3$ in Figure 1. In what follows, we will
canonically identify the hyperplane $\mathcal{G}$ with $\mathbb{R}^{d-1}$ by
formally dropping the last coordinate $\left\langle y,e_{d}\right\rangle=1$ of
points $y\in\mathcal{G}$. This allows us to write also
$\xi(x)\in\mathbb{R}^{d-1}$ for any
$x\in\mathbb{S}^{d-1}\setminus\mathbb{S}^{d-1}_{0}$. The map (9) is called the
_gnomonic projection_ of the sphere $\mathbb{S}^{d-1}$ into $\mathcal{G}$ (or
$\mathbb{R}^{d-1}$, see Besau and Werner,, 2016). It is a double covering of
$\mathbb{R}^{d-1}$, once for points from $\mathbb{S}^{d-1}_{+}$ and once for
those from $\mathbb{S}^{d-1}_{-}$. On $\mathbb{S}^{d-1}_{+}$ (or
$\mathbb{S}^{d-1}_{-}$) it is a bijection with $\mathbb{R}^{d-1}$. The points
from the equator $\mathbb{S}^{d-1}_{0}$ are not considered in (9). This will
not be a problem because of our assumption (8).

Figure 1. Transformation $\xi$ that takes
$\mathbb{S}^{d-1}\setminus\mathbb{S}^{d-1}_{0}$ to $\mathcal{G}$ for $d=2$
(left panel) and $d=3$ (right panel). The data from the northern hemisphere
$\mathbb{S}^{d-1}_{+}$ are mapped into points of positive $P_{\pm}$-mass in
$\mathcal{G}$ (red points), while data from the southern hemisphere
$\mathbb{S}^{d-1}_{-}$ project to points of negative $P_{\pm}$-mass (green
points). In the left panel we see also a halfspace $H_{0,u}\in\mathcal{H}_{0}$
(shaded halfplane). Only points $x_{1}\in\mathbb{S}^{1}_{+}$ and
$x_{2}\in\mathbb{S}^{1}_{-}$ lie in $H_{0,u}$, which means that precisely
$\xi(x_{1})$ and $\xi(x_{4})$ are contained in $G_{u}=H_{0,u}\cap\mathcal{G}$.
This is in accordance with formula (30).

Denote for any $B\subseteq\mathcal{G}$ (or $B\subseteq\mathbb{R}^{d-1}$) Borel

$\displaystyle P_{+}(B)$
$\displaystyle=P\left(\left\\{y\in\mathbb{S}^{d-1}_{+}\colon\xi(y)\in
B\right\\}\right),$ $\displaystyle P_{-}(B)$
$\displaystyle=P\left(\left\\{y\in\mathbb{S}^{d-1}_{-}\colon\xi(y)\in
B\right\\}\right),$

the pushforward measures of restrictions of $P$ to $\mathbb{S}^{d-1}_{+}$ and
$\mathbb{S}^{d-1}_{-}$, respectively, under the gnomonic projection $\xi$. In
words, $P_{+}$ is the image measure of the part of $P$ in the northern
hemisphere when projected to $\mathcal{G}$ (or $\mathbb{R}^{d-1}$), and
$P_{-}$ analogously for the southern hemisphere. We patch $P_{+}$ and $P_{-}$
together and use them to define a signed measure
$P_{\pm}\in\mathcal{S}\left({\mathbb{R}^{d-1}}\right)$ given by (4)

(10) $P_{\pm}(B)=P\left(\left\\{x\in\mathbb{S}^{d-1}_{+}\colon\xi(x)\in
B\right\\}\right)-P\left(\left\\{x\in\mathbb{S}^{d-1}_{-}\colon\xi(x)\in
B\right\\}\right)\quad\mbox{for $B\subseteq\mathbb{R}^{d-1}$ Borel}.$

The signed measure $P_{\pm}$ takes a simple form when $P$ is an empirical
measure of $n$ data points $x_{1},\dots,x_{n}\in\mathbb{S}^{d-1}$. In that
situation, $P_{\pm}$ is simply a signed empirical measure, with atoms of
positive mass $1/n$ at each $\xi(x_{i})$ such that
$x_{i}\in\mathbb{S}^{d-1}_{+}$, and atoms of negative mass $-1/n$ at
$\xi(x_{i})$ for $x_{i}\in\mathbb{S}^{d-1}_{-}$. In this situation, the
gnomonic projection was essential to design fast computational algorithms for
the sample ahD in Dyckerhoff and Nagy, (2023).

Our principal tool for analyzing ahD is its relation with hD, which is
described in the following theorem. In that result, we say that a set $H$ is
called a _generalized halfspace_ in $\mathbb{R}^{d-1}$ if $H$ is a halfspace
in $\mathbb{R}^{d-1}$, an empty set, or the whole space $\mathbb{R}^{d-1}$
itself.

######  Theorem 1.

Let $P\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$ and
$x\in\mathbb{S}^{d-1}$ be such that (8) is true. Then

(11)
$ahD(x;P)=P\left(\mathbb{S}^{d-1}_{-}\right)+\inf\left\\{P_{+}\left(H\right)-P_{-}\left(H^{\circ}\right)\colon
H\in\mathcal{H}^{*}\mbox{ and }\xi(x)\in H\right\\}$

for $\mathcal{H}^{*}$ the set of all generalized halfspaces in
$\mathbb{R}^{d-1}$. If, in addition,

(12) $P\left(\partial H\cap\mathbb{S}^{d-1}_{-}\right)=0\quad\mbox{for all
}H\in\mathcal{H}_{0},$

then

$ahD(x;P)=P(\mathbb{S}^{d-1}_{-})+hD(\xi(x);P_{\pm}).$

Formula (11) will be quite useful in deriving theoretical properties of ahD
and constructing examples, as we will see throughout this paper.

##  3\. Desiderata for a directional depth function

There are several well-established properties that a depth $D$ in a linear
space $\mathbb{R}^{d}$ should obey. It is typically agreed that $D$ should be
• affine invariant, meaning that it does not depend on the coordinate system
in $\mathbb{R}^{d}$;  • maximized at the center of symmetry of $P$ for any
$P\in\mathcal{P}\left({\mathbb{R}^{d}}\right)$ symmetric;  • monotonically
decreasing as $x$ moves along straight lines starting at the point of maximum
of $D$; and  • decaying uniformly to zero as $\left\|x\right\|\to\infty$.
This set of properties was postulated by Zuo and Serfling, (2000); for
additional related axioms we refer to Liu, (1990), Serfling, (2006), Mosler,
(2013), and Mosler and Mozharovskyi, (2022).

Compared to depths in $\mathbb{R}^{d}$, much less is known about general depth
functions defined in $\mathbb{S}^{d-1}$. One set of assumptions has been laid
down recently in Nagy et al., (2023). There, a general _angular_ (or
_directional_) _depth function_ $aD$ was introduced as a bounded map
$aD\colon\mathbb{S}^{d-1}\times\mathcal{P}\left({\mathbb{S}^{d-1}}\right)\to[0,\infty)$
that fulfills (most of) the following properties for all
$P\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$:

  1. ($\mathrm{D}_{1}$)

_Rotational invariance:_ $aD(x;P)=aD(Ox;P_{OX})$ for all
$x\in\mathbb{S}^{d-1}$ and any orthogonal matrix $O\in\mathbb{R}^{d\times d}$,
where $P_{OX}\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$ is the
distribution of the transformed random vector $OX$ with $X\sim P$;

  2. ($\mathrm{D}_{2}$)

_Maximality at center:_ For any
$P\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$ rotationally symmetric around
the axis given by $\mu\in\mathbb{S}^{d-1}$ we have

(13) $\max\\{aD(\mu;P),aD(-\mu;P)\\}=\sup_{x\in\mathbb{S}^{d-1}}aD(x;P);$

  3. ($\mathrm{D}_{3}$)

_Monotonicity along great circles:_

$aD(x;P)\leq
aD(\left(\mu+\alpha(x-\mu)\right)/\left\|\mu+\alpha(x-\mu)\right\|;P)$

for all $x\in\mathbb{S}^{d-1}\setminus\left\\{-\mu\right\\}$ and
$\alpha\in[0,1]$, where $\mu\in\mathbb{S}^{d-1}$ is any point that satisfies

(14) $aD(\mu;P)=\sup_{x\in\mathbb{S}^{d-1}}aD(x;P);$

  4. ($\mathrm{D}_{4}$)

_Minimality at the anti-median:_
$aD(-\mu;P)=\inf_{x\in\mathbb{S}^{d-1}}aD(x;P)$, for any
$\mu\in\mathbb{S}^{d-1}$ that satisfies (14).

  5. ($\mathrm{D}_{5}$)

_Upper semi-continuity:_ $aD(\cdot;P)\colon\mathbb{S}^{d-1}\to[0,\infty)\colon
x\mapsto aD(x;P)$ is upper semi-continuous, meaning that

${\lim\sup}_{x_{n}\to x}aD(x_{n};P)\leq aD(x;P)\quad\mbox{for all
}x\in\mathbb{S}^{d-1}$

where the sequence $\left\\{x_{n}\right\\}_{n=1}^{\infty}$ is taken in
$\mathbb{S}^{d-1}$.

  6. ($\mathrm{D}_{6}$)

_Quasi-concavity:_ All central regions

(15) $aD_{\alpha}(P)=\left\\{x\in\mathbb{S}^{d-1}\colon
aD(x;P)\geq\alpha\right\\}\quad\mbox{with $\alpha\geq 0$}$

are spherical convex sets.

  7. ($\mathrm{D}_{7}$)

_Non-rigidity of central regions:_ There exists a measure
$P\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$ such that for some $\alpha>0$
the central region from (15) of $aD$ is not a spherical cap.

Conditions ($\mathrm{D}_{1}$)–($\mathrm{D}_{4}$) are direct translations of
the classical requirements P1–P4 postulated for the (Euclidean) statistical
depth function in $\mathbb{R}^{d}$ in Zuo and Serfling, (2000). Analogues of
the additional conditions ($\mathrm{D}_{5}$) and ($\mathrm{D}_{6}$) have been
introduced in the analysis of the depth in $\mathbb{R}^{d}$ by Serfling,
(2006). The final condition ($\mathrm{D}_{7}$) appears in Nagy et al., (2023)
for the first time. It is a minimal requirement on an angular depth function
that guarantees that $aD$ reflects the shape properties of the distribution
$P\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$.

Condition ($\mathrm{D}_{2}$) operates with the notion of rotational symmetry
(Ley and Verdebout,, 2017, Section 2.3.2) of $P$. Recall that a distribution
$P\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$ is said to be _rotationally
symmetric_ around a direction $\mu\in\mathbb{S}^{d-1}$ if $X\sim P$ has the
same distribution as $OX$ for any orthogonal matrix $O\in\mathbb{R}^{d\times
d}$ that fixes $\mu$, that is $O\mu=\mu$. The center of rotational symmetry of
$P$ is never unique; if $\mu\in\mathbb{S}^{d-1}$ is a center of rotational
symmetry of $P$, then so is $-\mu\in\mathbb{S}^{d-1}$. Thus, the maximum on
the left-hand side of (13) is necessary to be considered in condition
($\mathrm{D}_{2}$).

The set of conditions ($\mathrm{D}_{1}$)–($\mathrm{D}_{7}$) is not
independent; clearly, ($\mathrm{D}_{6}$) is a stronger version of
($\mathrm{D}_{3}$). We list both these requirements since in Nagy et al.,
(2023), it was argued that the quasi-concavity condition ($\mathrm{D}_{6}$)
takes a quite different meaning on the unit sphere than it does in the
classical case of $\mathbb{R}^{d}$. In particular, in Nagy et al., (2023,
Theorem 3) it is proved that ($\mathrm{D}_{6}$) implies that $aD$ must be
constant on an open hemisphere in $\mathbb{S}^{d-1}$. Thus, for directional
data, the requirement ($\mathrm{D}_{6}$) of convexity of central regions is
questionable, and the strictly weaker ($\mathrm{D}_{3}$) may be preferable for
some depths. We will, however, see that just like the (Euclidean) hD in
$\mathbb{R}^{d}$, also ahD in $\mathbb{S}^{d-1}$ satisfies the stronger
condition ($\mathrm{D}_{6}$) with all its implications.

In the following subsections, we deal with conditions
($\mathrm{D}_{1}$)–($\mathrm{D}_{7}$) one by one, but not necessarily in this
order. We establish that each of our conditions is verified by ahD.

###  3.1. Rotational invariance

The validity of ($\mathrm{D}_{1}$) for $ahD$ was proved already in Small,
(1987, Example 4.4.4). It follows directly from the definition of $ahD$; for
completeness, we provide a proof.

######  Theorem 2.

The depth ahD satisfies condition ($\mathrm{D}_{1}$).

###### Proof.

Take $x\in\mathbb{S}^{d-1}$, and $O\in\mathbb{R}^{d\times d}$ orthogonal. To
find $ahD(Ox;P_{OX})$, we need to search through all halfspaces $H_{0,v}$ with
$v\in\mathbb{S}^{d-1}$ such that $Ox\in H_{0,v}$. Since $u\mapsto Ou$ is a
bijection of $\mathbb{S}^{d-1}$, we can equivalently write $v=Ou$ and search
over all $u\in\mathbb{S}^{d-1}$. The condition $Ox\in H_{0,v}=H_{0,Ou}$ then
translates to

(16) $0\leq\left\langle
Ox,Ou\right\rangle=(Ox)^{\mathsf{T}}(Ou)=x^{\mathsf{T}}O^{\mathsf{T}}Ou=x^{\mathsf{T}}u=\left\langle
x,u\right\rangle,$

i.e., it is equivalent with $x\in H_{0,u}$. Further, using (16) again we can
write

$P_{OX}(H_{0,Ou})=P\left(\left\\{y\in\mathbb{S}^{d-1}\colon Oy\in
H_{0,Ou}\right\\}\right)=P\left(\left\\{y\in\mathbb{S}^{d-1}\colon y\in
H_{0,u}\right\\}\right)=P(H_{0,u}),$

meaning that in both $ahD(x;P)$ and $ahD(Ox;P_{OX})$ one considers the same
collection of halfspaces. Necessarily, ($\mathrm{D}_{1}$) is true. ∎

The result of Small, (1987, Example 4.4.4) is, actually, stronger than
($\mathrm{D}_{1}$). It says that for any non-singular matrix
$A\in\mathbb{R}^{d\times d}$ and

(17) $\varphi_{A}\colon\mathbb{S}^{d-1}\to\mathbb{S}^{d-1}\colon
x\mapsto\frac{Ax}{\left\|Ax\right\|}$

the invariance $ahD(x;P)=ahD(\varphi_{A}(x);P_{\varphi_{A}(X)})$ holds true
for all $x\in\mathbb{S}^{d-1}$ and $X\sim
P\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$. The map (17) is a full-
dimensional linear transform $x\mapsto Ax$ in $\mathbb{R}^{d}$, followed by a
projection back to $\mathbb{S}^{d-1}$. It is more general than the rotations
considered in ($\mathrm{D}_{1}$); for $O\in\mathbb{R}^{d\times d}$ orthogonal,
we obtain $\varphi_{O}(x)=Ox$ by (16) and we recover ($\mathrm{D}_{1}$).
Unlike in ($\mathrm{D}_{1}$), maps (17) also allow “stretching” the sphere
$\mathbb{S}^{d-1}$ to an ellipsoid $A\,\mathbb{S}^{d-1}=\left\\{A\,x\colon
x\in\mathbb{S}^{d-1}\right\\}$ before mapping it back to itself.

###  3.2. (Semi-)Continuity with consequences

We now derive ($\mathrm{D}_{5}$) for ahD, but in doing so, we prove more: the
depth $ahD(x;P)$ is upper semi-continuous as a function of both arguments
$x\in\mathbb{S}^{d-1}$ and $P\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$.
For that result, we need to endow $\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$
with a topology; a natural one is the topology of weak convergence of
measures.

######  Theorem 3.

The angular halfspace depth (3) is upper semi-continuous as a function of
$\left(x,P\right)\in\mathbb{S}^{d-1}\times\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$,
meaning that

${\lim\sup}_{n\to\infty}ahD\left(x_{n};P_{n}\right)\leq ahD\left(x;P\right)$

whenever $x_{n}\to x$ in $\mathbb{S}^{d-1}$, and $P_{n}\xrightarrow{w}P$ in
$\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$ as $n\to\infty$. In particular,
ahD satisfies condition ($\mathrm{D}_{5}$). If, in addition, $P$ is smooth in
the sense that

(S) $P\left(\partial H\right)=0\mbox{ for all }H\in\mathcal{H}_{0},$

then ahD is continuous in both arguments, that is

$\lim_{n\to\infty}ahD\left(x_{n};P_{n}\right)=ahD\left(x;P\right)$

for $x_{n}\to x$ in $\mathbb{S}^{d-1}$, and $P_{n}\xrightarrow{w}P$ in
$\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$ as $n\to\infty$.

A simple consequence of the upper semi-continuity of the function (31) is that
the infimum in the definition of the depth (3) does not have to be attained.
Indeed, take $d=2$, and $P\in\mathcal{P}\left({\mathbb{S}^{1}}\right)$ given
as a mixture of the uniform distribution on the half-circle
$\left\\{x\in\mathbb{S}^{1}\colon\left\langle
x,e_{2}\right\rangle>0\right\\}=\mathbb{S}^{1}_{+}$ and an atom at
$\left(-1,0\right)^{\mathsf{T}}$ with equal weights $1/2$. Consider the
$ahD(x;P)$ of $x=\left(1,0\right)^{\mathsf{T}}$. Evidently,
$ahD\left(x;P\right)=0$ with $P\left(H_{0,u_{n}}\right)\to 0$ as
$u_{n}=\left(\cos(-\pi/2+1/n),\sin(-\pi/2+1/n)\right)^{\mathsf{T}}\to(0,-1)^{\mathsf{T}}$,
but no closed halfspace $H_{0,u}$ with $P\left(H_{0,u}\right)=0$ exists. The
problem with the non-existence of a minimizing halfspace $H_{0,u}$ satisfying
$ahD(x;P)=P(H_{0,u})$ can be resolved by considering so-called flag halfspaces
(Pokorný et al.,, 2023); we develop that theory in Section 3.4 below.

###  3.3. Quasi-concavity of level sets

Just as for hD, also ahD has convex upper level sets. In the following
theorem, we show a stronger claim: the level set

$ahD_{\alpha}(P)=\left\\{x\in\mathbb{S}^{d-1}\colon
ahD(x;P)\geq\alpha\right\\}$

can be written as an intersection of specific open hemispheres in
$\mathbb{S}^{d-1}$. Condition ($\mathrm{D}_{6}$) for $ahD$ follows immediately
since an intersection of (spherical) convex sets is always (spherical) convex.

######  Theorem 4.

For any $P\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$ and $\alpha\geq 0$ we
have

(18) $ahD_{\alpha}(P)=\bigcap\left\\{G\colon
G^{\mathsf{c}}\in\mathcal{H}_{0}\mbox{ and
}P(G^{\mathsf{c}})<\alpha\right\\}.$

In particular, ahD satisfies both ($\mathrm{D}_{3}$) and ($\mathrm{D}_{6}$).

Comparing Theorem 4 with the related result for hD from Rousseeuw and Ruts,
(1999, Proposition 6), we observe an intriguing discrepancy. While for hD the
upper level sets $hD_{\alpha}(P)$ can be written as intersections of _closed_
halfspaces whose complement has $P$-mass at most $\alpha$, in formula (18) we
used _open_ hemispheres. The following result shows that with closed
hemispheres in (18) we obtain only an inclusion.

######  Theorem 5.

For any $P\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$ and $\alpha\geq 0$ we
have

(19) $\bigcap\left\\{H\colon H\in\mathcal{H}_{0}\mbox{ and
}P(H^{\mathsf{c}})<\alpha\right\\}\subseteq ahD_{\alpha}(P).$

It is interesting to see that the opposite inclusion from (19) does not hold.
An example can be found in Appendix A.5. Formula (18) draws connections of ahD
with _spherical convex floating bodies_ studied in convex geometry (Besau and
Werner,, 2016). Indeed, for $P\in\mathbb{S}^{d-1}$ uniform on a spherical
convex body333A _spherical convex body_ is a closed spherically convex set
$K\subseteq\mathbb{S}^{d-1}$ such that $\mathrm{rad}\left(K\right)$ has non-
empty interior. $K\subset\mathbb{S}^{d-1}$ the spherical convex floating body
can be defined precisely as $ahD_{\alpha}(P)$ with appropriate $\alpha>0$
(Besau and Werner,, 2016, Definition 1). This observation parallels the
connections between classical (Euclidean) floating bodies and hD leveraged in
Nagy et al., (2019). A more detailed analysis of the relations of spherical
floating bodies with ahD can be conducted using tools from Laketa and Nagy,
(2022, Section 3).

###  3.4. Minimality at the anti-median and constancy on a hemisphere

The depth ahD has an interesting property observed first in Liu and Singh,
(1992, Proposition 4.6). For any distribution on a sphere, there exists a
hemisphere $H$ on which ahD is constant. For each $x\in H$, we then have that
$ahD(x;P)$ is equal to the minimum $P$-mass of a hemisphere in
$\mathbb{S}^{d-1}$. Especially in connection with our property
($\mathrm{D}_{4}$), it is important to note that this property does not hold
true for closed hemispheres.

######  Example 1.

Take $P\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$ a mixture of a uniform
distribution on $\mathbb{S}^{d-1}_{+}$ with weight $1/2$ and an atom of mass
$1/2$ at some point $z\in\mathbb{S}^{d-1}_{0}$. Then, $ahD(z;P)=1/2$ and
$ahD(x;P)=0$ for each
$x\in\left(\mathbb{S}^{d-1}_{-}\cup\mathbb{S}^{d-1}_{0}\right)\setminus\\{z\\}$.
For points $x\in\mathbb{S}^{d-1}_{+}$ we have $ahD(x;P)\in(0,1/2)$. Thus, the
unique angular halfspace median of $P$ is $z$. The infimum $P$-mass of a
(closed) hemisphere is $0$, but because of the point mass at $z$, no closed
hemisphere in $\mathbb{S}^{d-1}$ has constant null ahD. Still,
$ahD(-z;P)=0=\inf_{x\in\mathbb{S}^{d-1}}ahD(x;P)$, and condition
($\mathrm{D}_{4}$) is satisfied for this particular $P$.

The appropriate context to study the set of minimum ahD is that of flag
halfspaces, recently introduced in Pokorný et al., (2023). There, a slightly
more general version of the following definition can be found.

######  Definition.

Define $\mathcal{F}$ as the system of all sets $F$ of the form

(20) $F=\\{0\\}\cup\left(\bigcup_{k=1}^{d}G_{k}\right).$

Here,

  * •

$G_{d}\subset\mathbb{R}^{d}$ is an open halfspace whose boundary passes
through the origin.

  * •

For every $k=1,\dots,d-1$, the set $G_{k}$ is an open halfspace inside the
$k$-dimensional relative boundary of $G_{k+1}$, such that the relative
boundary of $G_{k}$ passes through the origin.

Any element of $\mathcal{F}$ is called a _flag halfspace_.

A flag halfspace in $\mathbb{R}^{3}$ is the union of  (i) an open halfspace
$G_{3}$ whose boundary passes through the origin,  (ii) a relatively open
halfplane $G_{2}$, inside the plane $\partial G_{3}$, whose relative boundary
passes through the origin, and  (iii) a ray $\\{0\\}\cup G_{1}$ from the
origin $0\in\mathbb{R}^{3}$ into one of the two directions in the line given
by the relative boundary of $G_{2}$.

Flag halfspaces are interesting because of their connections with hD. As
proved in Pokorný et al., (2023, Theorem 1), the infimum in the definition (5)
of hD can always be replaced by a minimum, if one searches through flag
halfspaces instead of closed halfspaces. An analogous result turns out to be
true also for ahD, as stated in the following theorem.

######  Theorem 6.

For any $P\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$ and
$x\in\mathbb{S}^{d-1}$ we have

$ahD(x;P)=\min\left\\{P(F)\colon F\in\mathcal{F}\mbox{ and }x\in F\right\\}.$

In particular, there always exists $F\in\mathcal{F}$ such that $x\in F$ and
$ahD(x;P)=P(F)$.

Armed with the notion of a flag halfspace, we now prove a sharp version of the
claim on hemispheres of minimum ahD.

######  Theorem 7.

Let $P\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$. Then there exists a flag
halfspace $F\in\mathcal{F}$ that satisfies

(21) $P(F)=\inf\left\\{P(G)\colon G\in\mathcal{F}\right\\}.$

For every $x\in F$ we then have
$ahD(x;P)=\min_{y\in\mathbb{S}^{d-1}}ahD(y;P)$. In particular,
($\mathrm{D}_{4}$) is true for ahD.

As a consequence of Theorem 7, we obtain much more than just condition
($\mathrm{D}_{4}$) for ahD. It holds true that for any $F\in\mathcal{F}$, we
have $x\in F\cap\mathbb{S}^{d-1}$ if and only if $-x\notin
F\cap\mathbb{S}^{d-1}$ (Laketa et al.,, 2022, Lemma 2.3). Thus, for any
$x\in\mathbb{S}^{d-1}$, at least one of the antipodal directions
$x,-x\in\mathbb{S}^{d-1}$ attains the minimum ahD

(22) $\min\\{ahD(x;P),ahD(-x;P)\\}=\min_{y\in\mathbb{S}^{d-1}}ahD(y;P).$

In our Example 1, for instance, we get that $F\in\mathcal{F}$ satisfying (21)
is any flag halfspace of the form (20) such that
$G_{d}\cap\mathbb{S}^{d-1}=\mathbb{S}^{d-1}_{-}$ and $z\notin G_{d-1}$. This
flag halfspace $F$ has null $P$-mass.

###  3.5. Maximality at the center

The following theorem states that condition ($\mathrm{D}_{2}$) is satisfied
for ahD.

######  Theorem 8.

Let $P\in\mathbb{S}^{d-1}$ be rotationally symmetric around
$\mu\in\mathbb{S}^{d-1}$. Then

$\max\left\\{ahD(\mu;P),ahD(-\mu;P)\right\\}=\max_{x\in\mathbb{S}^{d-1}}ahD(x;P),$

and ($\mathrm{D}_{2}$) is valid for ahD.

###  3.6. Conclusion: Desirable properties of angular depths

It remains to summarize our findings in Section 3: ahD verifies
($\mathrm{D}_{1}$) by Theorem 2; ($\mathrm{D}_{2}$) by Theorem 8;
($\mathrm{D}_{3}$) by Theorem 4; ($\mathrm{D}_{4}$) by Theorem 7;
($\mathrm{D}_{5}$) by Theorem 3; ($\mathrm{D}_{6}$) by Theorem 4; and
($\mathrm{D}_{7}$) because of Theorem 1. Overall, as argued in Nagy et al.,
(2023), it appears that ahD is the only angular depth function known in the
literature that verifies all conditions ($\mathrm{D}_{1}$)–($\mathrm{D}_{7}$).
This, of course, does not mean that ahD is in any sense an optimal depth. It,
however, hints that just as the classical hD in $\mathbb{R}^{d}$, also ahD in
$\mathbb{S}^{d-1}$ has the potential to be useful in many applications in
probability and statistics.

##  4\. Continuity and consistency properties

We now focus on continuity and consistency properties of ahD that are finer in
nature than the simple requirement ($\mathrm{D}_{5}$). In Section 4.1, we
treat the set of ahD-based directional medians and show that this set is
continuous as a set-valued mapping w.r.t. the topology of weak convergence in
$\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$. Then, in Section 4.2, we derive
a uniform continuity theorem for ahD in the argument of measure. In Section
4.3, we expand that theorem to the continuity of the central regions
$ahD_{\alpha}(P)$ from (18). Finally, we summarize and apply all our previous
advances to the sample ahD computed w.r.t. datasets in Section 4.4, which
gives remarkable strong uniform consistency properties for ahD.

To state our results, recall that for compact sets $K,L\subset\mathbb{R}^{d}$
is the _Hausdorff distance_ of $K$ and $L$ defined as

(23) $\delta_{H}(K,L)=\max\left\\{\sup_{x\in K}\inf_{y\in
L}\left\|x-y\right\|,\sup_{x\in L}\inf_{y\in K}\left\|x-y\right\|\right\\}.$

For closed sets $K,L\subseteq\mathbb{S}^{d-1}$ we simply embed
$\mathbb{S}^{d-1}$ into $\mathbb{R}^{d}$ canonically, and evaluate
$\delta_{H}(K,L)$ in $\mathbb{R}^{d}$ with $\left\|\cdot\right\|$ the
Euclidean distance. It would, of course, be possible to modify (23) to
$\mathbb{S}^{d-1}$ by considering directly the arc distance length instead of
the Euclidean distance. Thanks to the equivalence of all norms in finite-
dimensional spaces, the topology of this modification remains the same as for
$\delta_{H}$, and all our results thus hold true with both choices.

###  4.1. Properties of the angular halfspace median

We are concerned with the continuity properties of the ahD-based set of
directional medians, defined as the set of maximizers of ahD w.r.t.
$P\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$

(24)
$\mathfrak{M}(P)=ahD_{\alpha^{*}(P)}(P)=\left\\{x\in\mathbb{S}^{d-1}\colon
ahD\left(x;P\right)=\alpha^{*}(P)\right\\},$

with $\alpha^{*}(P)=\max_{y\in\mathbb{S}^{d-1}}ahD\left(y;P\right)$. By
Theorems 3 and 4, we know that $\mathfrak{M}(P)$ must be a non-empty compact
(spherical) convex set. The following theorem extends results from Donoho and
Gasko, (1992), Rousseeuw and Ruts, (1999), and Mizera and Volauf, (2002) to
the directional setup and ahD.

######  Theorem 9.

The following properties hold true:

  1. (i)

The maximum depth mapping

$\alpha^{*}\colon\mathcal{P}\left({\mathbb{S}^{d-1}}\right)\to\left[\frac{1}{d+1},1\right]\colon
P\mapsto\sup_{y\in\mathbb{S}^{d-1}}ahD\left(y;P\right)$

is upper semi-continuous.

  2. (ii)

At any $P\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$ that satisfies (S),
the mapping $\alpha^{*}$ is continuous and $\alpha^{*}(P)\leq 1/2$. Further,
the ahD-median mapping (24) is at $P$ an outer semi-continuous set-valued
mapping in the sense of Rockafellar and Wets, (1998, Definition 5.4). That is,
for any $P_{n}\xrightarrow{w}P$ as $n\to\infty$ in
$\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$ and $x_{n}\in\mathfrak{M}(P_{n})$
for each $n=1,2,\dots$, it holds true that all cluster points of the sequence
$\left\\{x_{n}\right\\}_{n=1}^{\infty}$ lie in $\mathfrak{M}(P)$.

  3. (iii)

Let $P\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$ satisfying (S) be such
that $\mathfrak{M}(P)$ is a singleton $\left\\{x\right\\}$. Take any
$P_{n}\xrightarrow{w}P$ as $n\to\infty$ in
$\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$ and
$x_{n}\in\mathfrak{M}(P_{n})$, $n=1,2,\dots$. Then there exists a sub-sequence
$\left\\{x_{n(k)}\right\\}_{k=1}^{\infty}$ of medians of $P_{n}$ such that
$x_{n(k)}\to x$ as $k\to\infty$. In particular, the ahD-median mapping (24) is
a continuous set-valued mapping in the sense of Rockafellar and Wets, (1998,
Definition 5.4), and also continuous in the sense of convergence in the
Hausdorff distance (23).

Note that in part (i) of Theorem 9, we also claim that the depth ahD of a
directional median on $\mathbb{S}^{d-1}$ cannot be lower than $1/(d+1)$. This
bound is attained, for instance, for the peculiar atomic distribution
$P\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$ described in the following
example.

######  Example 2.

Recall that $e_{j}\in\mathbb{S}^{d-1}$ is the $j$-th canonical vector, and
write
$e_{d+1}=\left(-1,\dots,-1\right)^{\mathsf{T}}/\sqrt{d}\in\mathbb{S}^{d-1}$.
Consider $P\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$ the uniform measure
supported in the set $\left\\{e_{1},\dots,e_{d},e_{d+1}\right\\}$. Then we
have $ahD\left(x;P\right)=1/(d+1)$ for all $x\in\mathbb{S}^{d-1}$. The proof
of this claim is in Appendix A.10.

The previous example is interesting because it demonstrates not only that
there exists a distribution with maximum ahD equal to the lower bound from
Theorem 9. Also, it shows that there is a distribution $X\sim
P\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$ which fails to be origin-
symmetric444A distribution $X\sim
P\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$ is said to be _origin
symmetric_ if $X$ and $-X$ have the same distribution., but its angular
halfspace depth is constant on the whole sphere.

Parts (ii) and (iii) of Theorem 9 were stated for hD in Mizera and Volauf,
(2002, Theorem 2). In the following example, we show that without the
assumption of the uniqueness of the ahD-median of $P$, we cannot guarantee the
continuity of the directional median set $\mathfrak{M}(P)$.

######  Example 3.

Consider first $Q\in\mathcal{P}\left({\mathbb{R}}\right)$ given as the mixture
of uniform distributions on the intervals $[-2,-1]$ and $[1,2]$, each with
weight $1/2$. The (standard Euclidean) halfspace median set of $Q$ is the
whole interval $[-1,1]$. Now, for $\varepsilon>0$ small and fixed, take
$Q_{n}$ assigning mass $1/2+\varepsilon/n$ to $[-2,-1]$ and mass
$1/2-\varepsilon/n$ to $[1,2]$. Certainly, $Q_{n}$ converges weakly to $Q$,
but the median set of $Q_{n}$ is
$\\{y_{n}\\}=\left\\{(n-4\varepsilon)/(n+2\varepsilon)\right\\}$, which is
contained inside the interval $[-2,-1]$. We see that the median mapping
$\mathfrak{M}$ for hD in $\mathbb{R}$ is outer semi-continuous but not
continuous at $Q$. To obtain corresponding directional distributions, we
project our setup to the upper semi-circle $\mathbb{S}^{1}_{+}$ of the circle
$\mathbb{S}^{1}$ using the inverse gnomonic projection from Section 2. Then,
we directly apply Theorem 1.

###  4.2. Continuity in measure

Under an appropriate smoothness condition (S), the mapping ahD can be shown to
be uniformly continuous w.r.t. the weak convergence of measures.

######  Theorem 10.

Suppose
$\left\\{P_{n}\right\\}_{n=1}^{\infty}\subset\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$
is a sequence of measures such that $P_{n}\xrightarrow{w}P$ as $n\to\infty$,
where $P\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$ satisfies the
smoothness condition (S). Then we can write

$\sup_{x\in\mathbb{S}^{d-1}}\left|ahD(x;P_{n})-ahD(x;P)\right|\xrightarrow[n\to\infty]{}0.$

Without the smoothness condition (S), we cannot assert that the uniform
continuity in Theorem 10 is true. This can be seen in an example where $P$ is
concentrated in $e_{d}\in\mathbb{S}^{d-1}$, but $P_{n}$ is uniform on a
spherical cap in $\mathbb{S}^{d-1}$ around $e_{d}$ with (spherical) radius
$1/n$. Then $P_{n}\xrightarrow[]{w}P$ as $n\to\infty$ and $ahD(e_{d};P)=1$,
but $ahD(e_{d};P_{n})=1/2$ for all $n$.

###  4.3. Continuity of the central regions

We now follow Dyckerhoff, (2017), who proved that under certain conditions,
the central regions $hD_{\alpha}(P)$ from (2) are continuous in the Hausdorff
distance as a function of $P\in\mathcal{P}\left({\mathbb{R}^{d}}\right)$. We
adapt those results from $\mathbb{R}^{d}$ to the setup of directional data in
$\mathbb{S}^{d-1}$ and ahD. For that, we will need a modification of the
strict monotonicity condition formulated in Dyckerhoff, (2017) for hD in
$\mathbb{R}^{d}$. We phrase a related requirement for ahD; we say that ahD is
_strictly monotone_ for $P\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$ if
for each
$\alpha\in\left(\min_{x\in\mathbb{S}^{d-1}}ahD(x;P),\max_{x\in\mathbb{S}^{d-1}}ahD(x;P)\right)$
we have

(25) $ahD_{\alpha}(P)=\mathrm{cl}\left(\left\\{x\in\mathbb{R}^{d}\colon
ahD(x;P)>\alpha\right\\}\right),$

where $\mathrm{cl}\left(A\right)$ stands for the closure of the set
$A\subseteq\mathbb{S}^{d-1}$. Roughly speaking, strict monotonicity means that
there are no regions of $\mathbb{S}^{d-1}$ of constant depth equal to
$\alpha>\min_{x\in\mathbb{S}^{d-1}}ahD(x;P)$. Of course, due to Theorem 7, we
have to exclude the hemisphere of minimum $P$-mass (i.e.,
$\alpha>\min_{x\in\mathbb{S}^{d-1}}ahD(x;P)$), since at
$\alpha=\min_{x\in\mathbb{S}^{d-1}}ahD(x;P)$ the condition (25) is never
satisfied.

######  Theorem 11.

Suppose that ahD is strictly monotone for
$P\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$, and that
$\left\\{P_{n}\right\\}_{n=1}^{\infty}\subset\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$
is a sequence of measures that satisfies

(26)
$\lim_{n\to\infty}\sup_{x\in\mathbb{S}^{d-1}}\left|ahD(x;P_{n})-ahD(x;P)\right|=0.$

Then for any closed interval
$A\subset(\min_{x\in\mathbb{S}^{d-1}}ahD(x;P),\max_{x\in\mathbb{S}^{d-1}}ahD(x;P))$
we can write

(27) $\lim_{n\to\infty}\sup_{\alpha\in
A}\delta_{H}\left(ahD_{\alpha}(P_{n}),ahD_{\alpha}(P)\right)=0.$

The condition (26) is satisfied if $P$ is smooth (that is, (S) is valid) by
Theorem 10. The additional condition (25) of strict monotonicity of
$P\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$ is more delicate. In the
setup of hD in $\mathbb{R}^{d}$, it was argued in Laketa and Nagy, (2022,
Section 4.3) that smoothness of $P\in\mathcal{P}\left({\mathbb{R}^{d}}\right)$
and the connectedness of its support already guarantee a variant of (25). In
$\mathbb{S}^{d-1}$, however, this is not enough, as we demonstrate in the
following example.

######  Example 4.

Take $X\sim P\in\mathcal{P}\left({\mathbb{S}^{1}}\right)$, where the random
variable $X\in\mathbb{S}^{1}$ is encoded by its angle $\theta\in(-\pi,\pi]$
with the positive first coordinate axis in $\mathbb{R}^{2}$. The measure $P$
is given as a mixture of four uniform distributions:  (i) of mass $1/2$ in the
angle $(\pi/2,\pi]$,  (ii) of mass $1/4$ in the angle $(0,\pi/4]$,  (iii) of
mass $1/8$ in the angle $[\pi/4,\pi/2)$, and  (iv) of mass $1/8$ in the angle
$[-3\pi/4,-\pi/2)$. See also Figure 2.

Figure 2. Example 4: A distribution
$P\in\mathcal{P}\left({\mathbb{S}^{1}}\right)$ that does not satisfy the
condition of strict monotonicity (25) at $\alpha=3/8$. In the left panel, the
outer red curve corresponds to the ahD of
$x(\theta)=(\cos(\theta),\sin(\theta))^{\mathsf{T}}\in\mathbb{S}^{1}$ as a
function of the angle $\theta\in(-\pi,\pi]$; the depth ahD is maximized at the
angle $\theta=5\pi/8$ (solid straight line), and $ahD_{3/8}(P)$ corresponds to
the angles $\theta\in[\pi/4,3\pi/4]$ (wedge with dashed boundary lines). The
depth ahD is constant for $\theta\in[\pi/4,\pi/2]$, meaning that ahD is not
strictly monotone for $P$. In the right panel, we have the function
$\theta\mapsto ahD(x(\theta);P)$ (blue curve).

The minimum $P$-mass of a hemisphere is $1/8$, and $ahD(x;P)=1/8$ for all
$x\in\mathbb{S}^{1}_{-}\cup\mathbb{S}^{1}_{0}$. For
$x=(\cos(\theta),\sin(\theta))^{\mathsf{T}}\in\mathbb{S}^{1}$ we get ahD

(28) $ahD(x;P)=\begin{cases}1/8&\mbox{for }\theta\in(-\pi,0],\\\
1/8+\theta/\pi&\mbox{for }\theta\in(0,\pi/4],\\\ 3/8&\mbox{for
}\theta\in(\pi/4,\pi/2],\\\
1/2-\frac{\left|\theta-5\pi/8\right|}{\pi}&\mbox{for
}\theta\in(\pi/2,3\pi/4],\\\ 1/8+\frac{\pi-\theta}{\pi}&\mbox{for
}\theta\in(3\pi/4,\pi].\end{cases}$

In particular, for $\alpha=3/8$ we have that $ahD_{\alpha}(P)$ corresponds to
the interval of angles $\theta\in[\pi/4,3\pi/4]$, but the (spherical) closure
of the set $\left\\{x\in\mathbb{S}^{1}\colon ahD(x;P)>3/8\right\\}$ is the
interval of angles $\theta\in[\pi/2,3\pi/4]$. The strict monotonicity
condition (25) is, therefore, not satisfied for $\alpha=3/8$.

Construct now for $\varepsilon>0$ small and fixed a sequence of measures
$\left\\{P_{n}\right\\}_{n=1}^{\infty}\subset\mathcal{P}\left({\mathbb{S}^{1}}\right)$
defined as similarly as $P$, but in the interval $\theta\in(\pi/2,\pi]$ we put
$P_{n}$-mass $1/2+(-1)^{n}\varepsilon$, and in the interval $(0,\pi/4]$ we
assign $P_{n}$-mass $1/4+(-1)^{n+1}\varepsilon$. In other words, for $n$ odd
we increase the mass in $(0,\pi/4]$ slightly above $1/4$ and decrease the mass
in $(\pi/2,\pi]$ below $1/2$, and for $n$ even the other way around. We
certainly have that $P_{n}$ converges weakly to $P$ as $n\to\infty$, and since
$P$ satisfies the smoothness condition (S), Theorem 10 also gives that
$\sup_{x\in\mathbb{S}^{1}}\left|ahD(x;P_{n})-ahD(x;P)\right|$ vanishes as
$n\to\infty$. Nevertheless, a simple computation as in (28) gives that for
$\alpha=3/8$ we have $ahD_{\alpha}(P_{n})$ contains the interval of angles
$\theta\in(\pi/4,\pi/2]$ for $n$ odd, but not for $n$ even. As such, the
sequence of sets $\left\\{ahD_{\alpha}(P_{n})\right\\}_{n=1}^{\infty}$ does
not converge in the Hausdorff distance to any set, and (27) cannot be true.

Note that the problem with Example 4 does not rest in the fact that the
support of $P$ is disconnected; one could easily add another mixture component
supported on $\mathbb{S}^{1}$ to $P$ with sufficiently low weight, and the
same phenomenon appears. The core of the problem is in the symmetry of $P$ in
the angle $\theta\in[\pi/4,\pi/2)$, giving that any halfspace
$H_{u}\in\mathcal{H}_{0}$ with $u=(\cos(\theta),\sin(\theta))^{\mathsf{T}}$
and $\theta\in[3\pi/4,\pi/2]$ has the same $P$-mass equal to $\alpha=3/8$. One
natural way of enforcing (25) is to forbid these symmetries, as done in the
following result.

######  Theorem 12.

Suppose that $P\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$ has a dominant
hemisphere, meaning that there exists $u\in\mathbb{S}^{d-1}$ such that for all
$B\subseteq\mathbb{S}^{d-1}\cap H_{u}$ of non-null spherical Lebesgue measure
we have $P(B)>P(-B)$. Assume, in addition, that $P$ is smooth in the sense of
(S). Then is the condition (25) satisfied for $P$.

Another way of obtaining the strict monotonicity condition (25) is to use
directly Theorem 1 and assume that $P$ is supported in a hemisphere $H$. Then,
we can assume that after gnomonic projection to the tangent hyperplane at the
pole of $H$, the (non-negative) measure $P_{\pm}$ in $\mathbb{R}^{d-1}$
satisfies smoothness and contiguity of its support as assumed in Laketa and
Nagy, (2022, Section 4.3).

###  4.4. Large sample properties

As our final result, we consider the problem of estimating ahD from a random
sample. We consider a sequence $X_{1},\dots,X_{n}\in\mathbb{S}^{d-1}$ of
independent random variables defined on $\Omega$ sampled from the distribution
$P\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$. We attach to each
$X_{i}\equiv X_{i}(\omega)$ mass $1/n$ and denote the resulting (random)
empirical measure by
$P_{n}(\omega)\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$. When computing
the depth, we typically estimate the depth of the (unknown) distribution $P$
by means of the sample depth, based on plugging the empirical measure
$P_{n}(\omega)$ into the depth (3) instead of $P$. The depth
$ahD(x;P_{n}(\omega))$ is called the sample angular halfspace depth of $x$. In
the following result, we prove that the sample ahD almost surely uniformly
approximates its population counterpart $ahD(\cdot;P)$ as $n\to\infty$, and
the same is true for the derived quantities $\mathfrak{M}(P_{n}(\omega))$ and
$ahD_{\alpha}(P_{n}(\omega))$.

######  Theorem 13.

Suppose that $X_{1},X_{2},\dots,$ is a sequence of independent random
variables with distribution $P\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$,
and let $P_{n}(\omega)\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$ be the
(random) empirical measure corresponding to
$X_{1}(\omega),\dots,X_{n}(\omega)$.

  1. (i)

Then we have

$\mathsf{P}\left(\left\\{\omega\in\Omega\colon\lim_{n\to\infty}\sup_{x\in\mathbb{S}^{d-1}}\left|ahD(x;P_{n}(\omega))-ahD(x;P)\right|=0\right\\}\right)=1.$

  2. (ii)

If $P\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$ is such that ahD is
strictly monotone for $P$, then for any closed interval
$A\subset(\min_{x\in\mathbb{S}^{d-1}}ahD(x;P),\max_{x\in\mathbb{S}^{d-1}}ahD(x;P))$
we can write

$\mathsf{P}\left(\left\\{\omega\in\Omega\colon\lim_{n\to\infty}\sup_{\alpha\in
A}\delta_{H}\left(ahD_{\alpha}(P_{n}(\omega)),ahD_{\alpha}(P)\right)=0\right\\}\right)=1.$

  3. (iii)

If $P\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$ satisfies the conditions
of part (iii) of Theorem 9, then

$\mathsf{P}\left(\left\\{\omega\in\Omega\colon\lim_{n\to\infty}\delta_{H}\left(\mathfrak{M}(P_{n}(\omega)),\mathfrak{M}(P)\right)=0\right\\}\right)=1.$

##  Appendix A Proofs of the theoretical results

###  A.1. Proof of Theorem 1

In the definition of ahD in (3) we only consider halfspaces in
$\mathbb{R}^{d}$ whose boundary passes through the origin. Denote by $G_{u}$
the intersection of $\mathcal{G}$ and $H_{0,u}\in\mathcal{H}_{0}$. When
considered as a subset of the affine space $\mathcal{G}$, the set $G_{u}$ is a
generalized halfspace:  (i) it is a halfspace in $\mathcal{G}$ if
$u\in\mathbb{S}^{d-1}\setminus\left\\{e_{d},-e_{d}\right\\}$,  (ii) an empty
set if $u=-e_{d}$, and  (iii) equals $\mathcal{G}$ if $u=e_{d}$. By the
definition (9) of the gnomonic projection $\xi$, a point
$y\in\mathbb{S}^{d-1}\setminus\mathbb{S}^{d-1}_{0}$ lies in $H_{0,u}$ if and
only if $\left\langle
y,u\right\rangle=\left\langle\xi(y),u\right\rangle\left\langle
y,e_{d}\right\rangle\geq 0$. If $y\in\mathbb{S}^{d-1}_{+}$, this is equivalent
with $\xi(y)\in G_{u}=H_{0,u}\cap\mathcal{G}$; for $y\in\mathbb{S}^{d-1}_{-}$
we have that $y\in H_{0,u}$ if and only if
$\left\langle\xi(y),u\right\rangle\leq 0$, that is $\xi(y)\in G_{-u}$. In
terms of $P_{\pm}$ we can thus express the $P$-mass of any
$H_{0,u}\in\mathcal{H}_{0}$ as

(29) $P\left(H_{0,u}\right)=P_{+}\left(G_{u}\right)+P_{-}\left(G_{-u}\right).$

Writing $G_{u}^{\circ}$ and $\partial G_{u}$ for the relative interior and the
relative boundary of the $(d-1)$-dimensional generalized halfspace $G_{u}$ in
$\mathcal{G}$, for any $u\in\mathbb{S}^{d-1}$ we have
$P\left(\mathbb{S}^{d-1}_{-}\right)=P_{-}\left(\mathcal{G}\right)=P_{-}\left(G_{-u}^{\circ}\right)+P_{-}\left(\partial
G_{-u}\right)+P_{-}\left(G_{u}^{\circ}\right)=P_{-}\left(G_{-u}\right)+P_{-}\left(G_{u}^{\circ}\right)$,
simply because the sets $G_{-u}$ and $G_{u}^{\circ}$ are disjoint and
decompose $\mathcal{G}$. Thus, we can rewrite (29) to

(30)
$P\left(H_{0,u}\right)=P\left(\mathbb{S}^{d-1}_{-}\right)+P_{+}\left(G_{u}\right)-P_{-}\left(G_{u}^{\circ}\right).$

This holds true for any normal vector $u\in\mathbb{S}^{d-1}$, including
$u=e_{d}$ or $-e_{d}$.

Since we assumed in (8) that $x\in\mathbb{S}^{d-1}_{+}$, we know that $x\in
H_{0,u}$ if and only if $\xi(x)\in G_{u}$. Plugging (30) into (3) we obtain

$\displaystyle ahD(x;P)$
$\displaystyle=P\left(\mathbb{S}^{d-1}_{-}\right)+\inf\left\\{P_{+}\left(G_{u}\right)-P_{-}\left(G_{u}^{\circ}\right)\colon
u\in\mathbb{S}^{d-1}\mbox{ and }x\in H_{0,u}\right\\}$
$\displaystyle=P\left(\mathbb{S}^{d-1}_{-}\right)+\inf\left\\{P_{+}\left(G_{u}\right)-P_{-}\left(G_{u}^{\circ}\right)\colon
u\in\mathbb{S}^{d-1}\mbox{ and }\xi(x)\in G_{u}\right\\}$
$\displaystyle=P\left(\mathbb{S}^{d-1}_{-}\right)+\inf\left\\{P_{+}\left(H\right)-P_{-}\left(H^{\circ}\right)\colon
H\in\mathcal{H}^{*}\mbox{ and }\xi(x)\in H\right\\}$

where in the last expression, it makes no difference whether the generalized
halfspaces $\mathcal{H}^{*}$ are considered in $\mathbb{R}^{d}$, or
$\mathcal{G}$, or $\mathbb{R}^{d-1}$ when identified with $\mathcal{G}$.
Considering $\mathcal{H}^{*}$ in $\mathbb{R}^{d-1}$, we have proved the first
part of our theorem.

Suppose now that $P_{-}$ satisfies the continuity condition (12). Then,
$P_{-}(\partial H)=0$ for all $H\in\mathcal{H}^{*}$ in $\mathbb{R}^{d-1}$ and
thus $P_{-}(H^{\circ})=P_{-}\left(H\right)$, which allows us to write
$P_{+}\left(H\right)-P_{-}\left(H^{\circ}\right)=P_{+}(H)-P_{-}(H)=P_{\pm}(H)$
in (11).

###  A.2. Proof of Theorem 3

We first use the portmanteau theorem (Dudley,, 2002, Theorem 11.1.1) to show
that the function

(31)
$f\colon\mathcal{P}\left({\mathbb{S}^{d-1}}\right)\times\mathbb{S}^{d-1}\to[0,1]\colon\left(P,u\right)\mapsto
P\left(H_{0,u}\right)$

is upper semi-continuous at any measure
$P\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$, and continuous at any $P$
that satisfies (S). To see that, fix
$P\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$ and take $u_{n}\to u$ in
$\mathbb{S}^{d-1}$. Since $u_{n}$ converges to $u$, it is possible to find a
sequence of orthogonal matrices
$\left\\{O_{n}\right\\}_{n=1}^{\infty}\subset\mathbb{R}^{d\times d}$ such that
$O_{n}u_{n}=u$ for each $n=1,2,\dots$, and $O_{n}$ converges to the identity
matrix $I_{d}\in\mathbb{R}^{d\times d}$. Define a measure
$Q_{n}\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$ as the distribution of
$O_{n}X_{n}$ with $X_{n}\sim P_{n}$. By Slutsky’s theorem (Jiang,, 2010,
Theorem 2.13), we then have that $Q_{n}\xrightarrow{w}P$ as $n\to\infty$. At
the same time, for each $n=1,2,\dots$ we can write

$O_{n}H_{0,u_{n}}=\left\\{O_{n}y\colon\left\langle y,u_{n}\right\rangle\geq
0\right\\}=\left\\{z\colon\left\langle
O_{n}^{\mathsf{T}}z,u_{n}\right\rangle\geq 0\right\\}=H_{0,O_{n}u_{n}},$

where we used $O_{n}^{-1}=O_{n}^{\mathsf{T}}$ as follows from the
orthogonality of $O_{n}$ and therefore, we have

(32) $P_{n}(H_{0,u_{n}})=\mathsf{P}\left(X_{n}\in
H_{0,u_{n}}\right)=\mathsf{P}\left(O_{n}X_{n}\in
O_{n}H_{0,u_{n}}\right)=Q_{n}(H_{0,u}).$

Thus, we have a fixed closed halfspace $H_{0,u}$ and a sequence of measures
$\left\\{Q_{n}\right\\}_{n=1}^{\infty}\subset\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$
such that $Q_{n}\xrightarrow{w}P$ as $n\to\infty$. The portmanteau theorem
(Dudley,, 2002, Theorem 11.1.1) and (32) then give that

${\lim\sup}_{n\to\infty}f(P_{n},u_{n})={\lim\sup}_{n\to\infty}P_{n}(H_{0,u_{n}})={\lim\sup}_{n\to\infty}Q_{n}(H_{0,u})\leq
P(H_{0,u})=f(P,u)$

as we wanted to show. In the special case when also (S) is valid, we obtain
that $H_{0,u}$ is a continuity set of $P$, and the portmanteau theorem gives
even

${\lim}_{n\to\infty}f(P_{n},u_{n})={\lim}_{n\to\infty}P_{n}(H_{0,u_{n}})={\lim}_{n\to\infty}Q_{n}(H_{0,u})=P(H_{0,u})=f(P,u).$

Now for $x\in\mathbb{S}^{d-1}$ denote

(33) $v(x)=\left\\{u\in\mathbb{S}^{d-1}\colon x\in
H_{0,u}\right\\}=\left\\{u\in\mathbb{S}^{d-1}\colon\left\langle
x,u\right\rangle\geq 0\right\\}=\mathbb{S}^{d-1}\cap H_{0,x}$

the set of all inner normals of halfspaces taken in the infimum in the
definition (3) of $ahD(x;P)$. The set-valued mapping $v(x)$ is continuous in
$x$ in the sense of Painlevé-Kuratowski convergence (Rockafellar and Wets,,
1998, Chapter 5.B), and ahD can be written as

$ahD\left(x;P\right)=\inf_{u\in v(x)}f(P,u).$

We have verified all the assumptions of Berge’s maximum theorem (Berge,, 1997,
pp. 115–117) on parametric optimization, which asserts that the depth function
$ahD\colon\mathbb{S}^{d-1}\times\mathcal{P}\left({\mathbb{S}^{d-1}}\right)\to[0,1]\colon(x,P)\mapsto
ahD(x;P)$ is upper semi-continuous in both arguments for general $P$, and
continuous in both arguments at any $P$ satisfying (S).

###  A.3. Proof of Theorem 4

Suppose first that $x\in ahD_{\alpha}(P)$, meaning that
$ahD\left(x;P\right)\geq\alpha$. If $x$ does not belong to the right-hand side
of (18), then there must exist an open halfspace $G$ whose boundary passes
through the origin, $P(G^{\mathsf{c}})<\alpha$, and $x\notin G$. Take
$H=G^{\mathsf{c}}\in\mathcal{H}_{0}$. Now, we have $x\in H\in\mathcal{H}_{0}$
and $P(H)<\alpha$, which contradicts $ahD\left(x;P\right)\geq\alpha$.
Necessarily, $ahD_{\alpha}(P)\subseteq\bigcap\left\\{G\colon
G^{\mathsf{c}}\in\mathcal{H}_{0}\mbox{ and
}P(G^{\mathsf{c}})<\alpha\right\\}$.

For the other direction, suppose that $x\notin ahD_{\alpha}(P)$. Then we have
$\alpha>ahD(x;P)=\inf_{H\in\mathcal{H}_{0}\colon x\in H}P(H)$, meaning that
there exists a closed halfspace $H\in\mathcal{H}_{0}$ such that $x\in H$ and
$P(H)<\alpha$. Consider the open halfspace $G=H^{\mathsf{c}}$. Then $x\notin
G$, $G^{\mathsf{c}}\in\mathcal{H}_{0}$ and $P(G^{\mathsf{c}})=P(H)<\alpha$,
meaning that $x$ does not belong to the right-hand side of (18). This gives
$ahD_{\alpha}(P)\supseteq\bigcap\left\\{G\colon
G^{\mathsf{c}}\in\mathcal{H}_{0}\mbox{ and
}P(G^{\mathsf{c}})<\alpha\right\\}$, and proves our claim.

###  A.4. Proof of Theorem 5

Take any $x\in\mathbb{S}^{d-1}$ that is not in $ahD_{\alpha}(P)$. Then
$ahD\left(x;P\right)<\alpha$ and therefore, there is a closed halfspace
$H\in\mathcal{H}_{0}$ such that $x\in H$ and $P(H)<\alpha$. First, we show
that we can also assume that $H$ satisfies the additional property $x\in
H^{\circ}$. Indeed, if $x\in\partial H$ and $u\in\mathbb{S}^{d-1}$ is the
inner normal of $H$, we choose $u_{n}=\cos(1/n)u+\sin(1/n)x$ for
$n=1,2,\dots$. Since $\left\langle x,u\right\rangle=0$ we have

$\left\|u_{n}\right\|^{2}=(\cos(1/n))^{2}\left\langle
u,u\right\rangle+(\sin(1/n))^{2}\left\langle x,x\right\rangle=1,$

and thus $u_{n}\in\mathbb{S}^{d-1}$. In addition,

$\left\langle x,u_{n}\right\rangle=\cos(1/n)\left\langle
x,u\right\rangle+\sin(1/n)\left\langle x,x\right\rangle=\sin(1/n)>0,$

and for $H_{n}=H_{0,u_{n}}$ we can write $x\in H_{n}^{\circ}$ for all $n$.
Halfspaces $H_{n}\in\mathcal{H}_{0}$ correspond to slightly tilted $H$ so that
$x$ is moved to the interior of $H_{n}$. In the proof of Theorem 3, we proved
that the function (31) given by $u\mapsto P(H_{0,u})$ is upper semi-
continuous. Since $u_{n}\to u$, we thus have

$\limsup_{n\rightarrow\infty}P(H_{n})\leq P(H)<\alpha.$

Necessarily, there must exist $n=1,2,\dots$ such that $P(H_{n})<\alpha$. For
such $n$ we have $x\in H_{n}^{\circ}$ as we wanted to show.

Take now this $H\in\mathcal{H}_{0}$ with the properties $x\in H^{\circ}$ and
$P(H)<\alpha$, and denote by $\tilde{H}=(H^{\circ})^{\mathsf{c}}$ the
complementary closed halfspace. Then $x\notin\tilde{H}$, but
$\tilde{H}\in\mathcal{H}_{0}$ and $P(\tilde{H}^{\mathsf{c}})=P(H^{\circ})\leq
P(H)<\alpha$. Thus, $x$ does not belong to the left-hand side of (19).

###  A.5. Example to Theorems 4 and 5

It is convenient to define $P\in\mathcal{P}\left({\mathbb{S}^{2}}\right)$ by
means of its gnomonic projection
$P_{\pm}\in\mathcal{S}\left({\mathbb{R}^{2}}\right)$ from (10). We then
project $P_{\pm}$ to $\mathbb{S}^{2}$ to obtain $P$, and use formula (11) to
evaluate ahD. For that we simplify the notation and write for
$y\in\mathbb{R}^{2}$

$shD\left(y;P_{\pm}\right)=\inf\left\\{P_{\pm}\left(H\right)+P_{-}\left(\partial
H\right)\colon H\in\mathcal{H}^{*}\mbox{ and }y\in H\right\\},$

giving by (11) that
$ahD(x;P)=P\left(\mathbb{S}^{d-1}_{-}\right)+shD\left(\xi(x);P_{\pm}\right)$.

We take the signed measure
$P_{\pm}\in\mathcal{S}\left({\mathbb{R}^{2}}\right)$ with three atoms, of
$P_{\pm}$-mass $1/5$ each, at points $y_{1}=(-1,0)^{\mathsf{T}}$,
$y_{2}=(0,0)^{\mathsf{T}}$ and $y_{3}=(1,0)^{\mathsf{T}}$, and two atoms, of
mass $-1/5$ each, at points $z_{1}=(0,-1)^{\mathsf{T}}$ and
$z_{2}=(0,1)^{\mathsf{T}}$. The total $P_{\pm}$-mass of $\mathbb{R}^{2}$ is
$1/5$; any closed halfspace $H\subset\mathbb{R}^{2}$ that contains both
$z_{1}$ and $z_{2}$ must also contain $y_{2}$. Thus, no closed halfspace in
$\mathbb{R}^{2}$ can have $P_{\pm}$-mass smaller than $-1/5$, and
$shD(y;P_{\pm})\geq-1/5$ for all $y\in\mathbb{R}^{2}$. A halfspace $H$ with
$P_{\pm}(H)=-1/5$ and $y\in H$ that does not contain any atoms of $P_{\pm}$ in
$\partial H$ can be found for any $y\notin A$, where $A$ is the closed line
segment between $y_{1}$ and $y_{3}$. Thus, $shD(y;P_{\pm})=-1/5$ for all
$y\notin A$. For any $y\in A$, we can always find a halfspace $H$ with
$P_{\pm}(H)=0$ and $y\in H$. We have that $shD(y;P_{\pm})=0$ for $y\in A$.

Figure 3. The setup from Appendix A.5: The five points on the sphere
$\mathbb{S}^{2}$ (three black points in $\mathbb{S}^{2}_{+}$ and two orange
points in $\mathbb{S}^{2}_{-}$; the origin is displayed in red. The depth ahD
of the corresponding measure is equal to $1/5$ for all $x\notin B$ and $2/5$
for all $x\in B$, where $B$ is the arc displayed in brown. The plane on the
left-hand side passing through the origin separates $\mathbb{R}^{3}$ into two
open halfspaces, each containing only a single black point. In the right-hand
panel, we see the same setup in the gnomonic projection in the plane
$\mathbb{R}^{2}$. The pink halfplane $G_{1}$ contains $P_{\pm}$-mass $0$,
while the orange one $G_{2}$ contains $P_{\pm}$-mass $-1/5$.

We now use the inverse gnomonic projection and formula (11) to transfer our
setup from $\mathbb{R}^{2}$ to $\mathbb{S}^{2}$. The points $y_{1}$, $y_{2}$,
$y_{3}$ map to $x_{1}=(-1/\sqrt{2},0,1/\sqrt{2})^{\mathsf{T}}$,
$x_{2}=(0,0,1)^{\mathsf{T}}$, $x_{3}=(1/\sqrt{2},0,1/\sqrt{2})^{\mathsf{T}}$
in $\mathbb{S}^{2}_{+}$, while $z_{1}$ and $z_{2}$ map to
$x_{4}=(0,-1/\sqrt{2},-1/\sqrt{2})^{\mathsf{T}}$ and
$(0,1/\sqrt{2},-1/\sqrt{2})^{\mathsf{T}}$, respectively. Our atomic measure
$P\in\mathcal{P}\left({\mathbb{S}^{2}}\right)$ has five atoms $x_{i}$,
$i=1,\dots,5$, each of $P$-mass $1/5$. Using Theorem 1 we get that
$ahD(x;P)=1/5$ for all $x\in\mathbb{S}^{2}\setminus B$ and $ahD(x;P)=2/5$ for
$x\in B$, where
$B=\left\\{\cos(\alpha)x_{1}+\sin(\alpha)x_{3}\colon\alpha\in[0,\pi/2]\right\\}$
is the shorter arc between $x_{1}$ and $x_{3}$. We obtain $ahD_{2/5}(P)=B$.
Take now a closed halfspace $H=H_{0,u}\in\mathcal{H}_{0}$ with
$u=(1,0,0)^{\mathsf{T}}$. We have $P(H^{\mathsf{c}})=P(\\{x_{1}\\})=1/5<2/5$,
but $H\cap ahD_{2/5}(P)$ is one half of the arc $B$. Thus, for $\alpha=2/5$
and, say, $x_{1}\in ahD_{\alpha}(P)$ we have $H\in\mathcal{H}_{0}$ that
satisfies $P(H^{\mathsf{c}})<\alpha$, but $x_{1}\notin H$. Therefore, the
inequality in (19) is strict.

###  A.6. Proof of Theorem 6

The proof is a direct adaptation of the proof of Pokorný et al., (2023,
Theorem 1).

###  A.7. Proof of Theorem 7

The first part of the statement follows directly from Pokorný et al., (2023,
Theorem 1). Indeed, the right-hand side of (21) is by that theorem precisely
the (Euclidean) halfspace depth of the origin $0\in\mathbb{R}^{d}$ w.r.t. the
measure $P$ when considered in the ambient space $\mathbb{R}^{d}$. Theorem 1
in Pokorný et al., (2023) then states that $hD(0;P)$ can also be expressed in
terms of our flag halfspaces (20), and a flag halfspace satisfying (21) always
exists.

For the second part of our claim, note that since by Theorem 6 also ahD can be
expressed using flag halfspaces, for every $x\in F$ from (21), we must have
$ahD(x;P)=P(F)$, directly because of (21) and the definition (3) of ahD.

###  A.8. Proof of Theorem 8

For any orthogonal matrix $O\in\mathbb{R}^{d\times d}$ such that $O\mu=\mu$ we
know, by the assumption of rotational symmetry of $P$, that $P_{OX}=P$. Thanks
to the rotational invariance ($\mathrm{D}_{1}$) of ahD from Theorem 2 we get

(34) $ahD(x;P)=ahD(Ox;P_{OX})=ahD(Ox;P).$

Thus, ahD of $x\in\mathbb{S}^{d-1}$ must be constant on spheres that are cut
from $\mathbb{S}^{d-1}$ by hyperplanes orthogonal to $\mu$. In other words,
$ahD(x;P)$ depends only on $\left\langle x,\mu\right\rangle$.

By Theorem 7, we know that (at least) one of the points
$\mu,-\mu\in\mathbb{S}^{d-1}$ lies in the region of minimum ahD of $P$.
Without loss of generality, let $-\mu$ be this point, meaning that

(35) $ahD(-\mu;P)=\min_{x\in\mathbb{S}^{d-1}}ahD(x;P).$

We will show that $ahD(\mu;P)=\max_{x\in\mathbb{S}^{d-1}}ahD(x;P)$. Suppose
for contradiction that there exists
$y\in\mathbb{S}^{d-1}\setminus\left\\{\mu,-\mu\right\\}$ such that

(36) $\alpha=ahD(y;P)>ahD(\mu;P)\geq ahD(-\mu;P),$

the second inequality following trivially from (35). In that case, formula
(34) gives that $ahD(x;P)=\alpha$ for all points $x$ on the set
$S=\left\\{x\in\mathbb{S}^{d-1}\colon\left\langle
x,\mu\right\rangle=\left\langle y,\mu\right\rangle\right\\}$. Necessarily,
$ahD_{\alpha}(P)\supseteq S$ but $\mu,-\mu\notin ahD_{\alpha}(P)$. By Theorem
4, the set $ahD_{\alpha}(P)$ must be spherically convex, meaning that the
convex hull of $\mathrm{rad}\left(S\right)$ intersected with
$\mathbb{S}^{d-1}$ is a subset of $ahD_{\alpha}(P)$. Now, if $\left\langle
y,\mu\right\rangle\neq 0$, the convex hull of $\mathrm{rad}\left(S\right)$
necessarily contains either $\mu$ or $-\mu$, which would contradict (36). The
only remaining case is that $\left\langle y,\mu\right\rangle=0$. That is,
however, also impossible thanks to our Theorem 7 that entails (22). Indeed, if
for some $y\in S$ we had (36), then (22) gives that
$ahD(-y;P)=ahD(-\mu;P)<\alpha$, but at the same time $-y\in S$ gives
$ahD(-y;P)=\alpha$, a contradiction. We have found that one of the points
$\mu$ or $-\mu$ must maximize the angular halfspace depth of $P$ as we wanted
to show.

###  A.9. Proof of Theorem 9

The following auxiliary lemma will be useful.

######  Lemma 14.

Let $U\subset\mathbb{S}^{d-1}$ be a set that is either open or finite in the
sphere $\mathbb{S}^{d-1}$. The following are equivalent:

  1. (A)

the origin lies in the convex hull of $U$;

  2. (B)

the origin lies in the convex hull of $R=\mathrm{rad}\left(U\right)$;

  3. (C)

$\bigcup_{r\in R}H_{0,r}=\mathbb{S}^{d-1}$;

  4. (D)

$\bigcup_{u\in U}H_{0,u}=\mathbb{S}^{d-1}$.

###### Proof of Lemma 14.

Since $U\subset R$, the convex hull of $U$ is a subset of the convex hull of
$R$, and (A) implies (B). For the opposite implication, suppose that the
origin lies in the convex hull of $R$. Carathéodory’s theorem (Schneider,,
2014, Theorem 1.1.4) gives that there exists $d+1$ (not necessarily distinct)
points $r_{1},\dots,r_{d+1}\in R$ such that

(37) $0=\sum_{i=1}^{d+1}\lambda_{i}\,r_{i}\mbox{ for some $0\leq\lambda_{i}$
with $\sum_{i=1}^{d+1}\lambda_{i}=1$.}$

Since $R=\mathrm{rad}\left(U\right)$, being a radial extension, never contains
the origin $0\in\mathbb{R}^{d}$, we have that
$\sum_{i=1}^{d+1}\lambda_{i}\left\|r_{i}\right\|>0$. From (37) we see that

$\sum_{i=1}^{d+1}\frac{\lambda_{i}\left\|r_{i}\right\|}{\sum_{j=1}^{d+1}\lambda_{j}\left\|r_{j}\right\|}\frac{r_{i}}{\left\|r_{i}\right\|}=0,$

and setting $u_{i}=r_{i}/\left\|r_{i}\right\|\in U$ and
$\gamma_{i}=\lambda_{i}\left\|r_{i}\right\|/\left(\sum_{j=1}^{d+1}\lambda_{j}\left\|r_{j}\right\|\right)$
yields that the origin is also contained in the convex hull of $U$. Therefore
(B) implies (A).

Statements (C) and (D) are clearly equivalent since $H_{0,r}=H_{0,u}$ for each
$r\in R$ and $u=r/\left\|r\right\|$.

Note that if $U$ is supposed to be open in the sphere, then the set $R$ from
(B) is open in $\mathbb{R}^{d}$. By Schneider, (2014, Theorem 1.1.10), we then
know that the convex hull $C$ of $R$ is open in $\mathbb{R}^{d}$. Suppose
first that (B) is not true. The Hahn-Banach theorem (Schneider,, 2014, Theorem
1.3.4) gives that there exists a vector $x\in\mathbb{S}^{d-1}$ such that
$H_{0,x}\cap C=\emptyset$, or equivalently $\left\langle x,r\right\rangle<0$
for all $r\in C$. The strict inequality follows either from the fact that $C$
is open (in the case of $U$ open), or from the finiteness of $U$. The
inequality $\left\langle x,r\right\rangle<0$ however translates into $x\notin
H_{0,r}$ for all $r\in R$, which gives that also (C) cannot be true.

Suppose finally that (C) is violated. Then there exists $x\in\mathbb{S}^{d-1}$
that is not contained in any $H_{0,r}$, or equivalently $\left\langle
x,r\right\rangle<0$ for all $r\in R$. If the origin was contained in the
convex hull $C$ of $R$, Schneider, (2014, Theorem 1.1.4) again gives that we
could write (37) for some $r_{1},\dots,r_{d+1}\in R$. The inner product of
(37) with $x$ then gives $0=\sum_{i=1}^{d+1}\lambda_{i}\,\left\langle
x,r_{i}\right\rangle<0$, a contradiction. Thus, (B) implies (C), and the lemma
is proved. ∎

Part (i). We begin by showing that the depth $\alpha^{*}(P)$ of an ahD-median
cannot be less than $1/(d+1)$. Suppose for contradiction that for some
$P\in\mathcal{P}\left({\mathbb{S}^{d-1}}\right)$ we have
$\alpha^{*}(P)<c<1/(d+1)$ for $c>0$. Then, for any $x\in\mathbb{S}^{d-1}$
there exists a halfspace $H_{0,u(x)}\in\mathcal{H}_{0}$ that contains $x$, and
$P(H_{0,u(x)})<c$. In other words, there is a covering of $\mathbb{S}^{d-1}$
by halfspaces whose $P$-probability is less than $c$. Consider the collection
of all inner normals $U=\left\\{u(x)\right\\}_{x\in\mathbb{S}^{d-1}}$ of such
halfspaces. By the upper semi-continuity of function $f$ from (31), the lower
level set $U$ of $f$ is open in $\mathbb{S}^{d-1}$ (Rockafellar and Wets,,
1998, Theorem 1.6). The auxiliary Lemma 14 below then gives that the origin
must be contained in the convex hull of $U$. Carathéodory’s theorem
(Schneider,, 2014, Theorem 1.1.4), in turn, gives that in that case, the
origin must be contained in a convex hull of at most $d+1$ (not necessarily
distinct) elements of the set $U$ denoted by
$u_{1},\dots,u_{d+1}\in\mathbb{S}^{d-1}$. Using once again Lemma 14, we have
that these $d+1$ elements induce a covering of $\mathbb{R}^{d}$, that is
$\bigcup_{i=1}^{d+1}H_{0,u_{i}}=\mathbb{R}^{d}$ with
$P\left(H_{0,u_{i}}\right)<c$ for all $i=1,\dots,d+1$. That means

$1=P\left(\mathbb{R}^{d}\right)=P\left(\bigcup_{i=1}^{d+1}H_{0,u_{i}}\right)\leq\sum_{i=1}^{d+1}P\left(H_{0,u_{i}}\right)\leq(d+1)c<1,$

a contradiction.

For the upper semi-continuity of the maximum depth mapping $\alpha^{*}$,
Theorem 3 gives that the function $ahD\left(\cdot;\cdot\right)$ is upper semi-
continuous in both arguments. Berge’s maximum theorem (Berge,, 1997, Theorem 2
on p. 116) then directly yields the conclusion.

Part (ii). Under the smoothness assumption (S), Theorem 3 gives that
$ahD\left(\cdot;\cdot\right)$ is continuous at $P$. Berge, (1997, Maximum
theorem on p. 117) then entails the continuity of $\alpha^{*}$, and the outer
semi-continuity of $\mathfrak{M}$.

Now, let $x\in\mathfrak{M}(P)$. Take any hyperplane $G$ containing $x$ and the
origin in $\mathbb{R}^{d}$, and consider the two halfspaces
$H,H^{\prime}\in\mathcal{H}_{0}$ determined by $G$. Because $P$ is smooth, we
have $P(G)=0$ and consequently $P(H)+P(H^{\prime})=1$, which gives
$\alpha^{*}(P)=ahD(x;P)\leq\min\left\\{P(H),1-P(H)\right\\}\leq 1/2$.

Part (iii). By Theorem 3 we know that $x_{n}\in\mathfrak{M}(P_{n})$ always
exists. Since $\mathbb{S}^{d-1}$ is a compact set, there exists a sub-sequence
$\left\\{x_{n(k)}\right\\}_{k=1}^{\infty}$ of that sequence of directional
medians such that $x_{n(k)}$ converges in $\mathbb{S}^{d-1}$ as $k\to\infty$.
Thanks to part (ii) of this theorem, we know that $x_{n(k)}$ then must
converge to $x$, the unique median of $P$. We just proved the inner semi-
continuity of the median mapping (24), which together with its outer semi-
continuity from part (ii) of this theorem gives the continuity of (24) in the
sense of Rockafellar and Wets, (1998, Definition 5.4). For a sequence of
closed sets $\left\\{K_{n}\right\\}_{n=1}^{\infty}$ that are all contained in
a bounded subset of $\mathbb{R}^{d}$, the convergence in the Hausdorff
distance $\lim_{n\to\infty}\delta_{H}\left(K_{n},K\right)=0$ is equivalent
with the convergence of sets induced by the notion of continuity of set-valued
mappings discussed above (Rockafellar and Wets,, 1998, Chapter 4C and p. 144).

###  A.10. Proof of Example 2

First note that since $0=\sum_{i=1}^{d+1}e_{i}/(d+1)$, Lemma 14 gives that the
collection of halfspaces $H_{0,e_{i}}$, $i=1,\dots,d+1$, covers
$\mathbb{S}^{d-1}$. In particular, for any $x\in\mathbb{S}^{d-1}$ there must
exist $i=1,\dots,d+1$ such that $\left\langle x,e_{i}\right\rangle\geq 0$,
implying $e_{i}\in v(x)$ for $v(x)=\left\\{u\in\mathbb{S}^{d-1}\colon x\in
H_{0,u}\right\\}=\mathbb{S}^{d-1}\cap H_{0,x}$, see also (33). Thus, any
halfspace $H_{0,u}\in\mathcal{H}_{0}$ with $u\in v(x)$ contains at least one
atom of $P$, and $ahD\left(x;P\right)\geq 1/(d+1)$. We now construct a
halfspace in $\mathcal{H}_{0}$ that contains $x$ and exactly one atom of $P$.
Since $x=\left(x_{1},\dots,x_{d}\right)^{\mathsf{T}}\in\mathbb{S}^{d-1}$, two
situations are possible:  (i) either all coordinates of $x$ are non-positive,
or  (ii) for some $j=1,\dots,d$ the coordinate $x_{j}$ is strictly positive.

In the first case, we consider the halfspace $H_{0,e_{d+1}}$. Then,
$\left\langle x,e_{d+1}\right\rangle>0$, i.e. $x\in H_{0,e_{d+1}}$, and at the
same time $\left\langle e_{i},e_{d+1}\right\rangle=-1$ for any $i\neq d+1$,
meaning that $e_{i}\notin H_{0,e_{d+1}}$. We get that $H_{0,e_{d+1}}$ contains
$x$ and $e_{d+1}$, but no other atom of $P$.

In the second case, let $j=1,\dots,d$ be the first index with $x_{j}>0$. Take
$\varepsilon>0$ small enough, and consider
$u=e_{j}-\left(\varepsilon,\dots,\varepsilon\right)^{\mathsf{T}}$ and the
halfspace $H_{0,u/\left\|u\right\|}$. A simple calculation gives

$\displaystyle\left\langle e_{j},u\right\rangle$
$\displaystyle=1-\varepsilon,$ $\displaystyle\left\langle
e_{i},u\right\rangle$ $\displaystyle=-\varepsilon,\mbox{ for $i=1,\dots,d$,
$i\neq j$},$ $\displaystyle\left\langle x,u\right\rangle$
$\displaystyle=x_{j}(1-\varepsilon)-\varepsilon\sum_{i\neq j}x_{i},$
$\displaystyle\mbox{ and }\quad\sqrt{d}\left\langle e_{d+1},u\right\rangle$
$\displaystyle=\varepsilon(d-1)-(1-\varepsilon),$

meaning that for $\varepsilon>0$ small enough, $x,e_{j}\in
H_{0,u/\left\|u\right\|}$, but $e_{i}\notin H_{0,u/\left\|u\right\|}$ for all
$i=1,\dots,d+1$, $i\neq j$. Thus, $H_{0,u/\left\|u\right\|}$ contains only $x$
and a single atom of $P$, meaning that $ahD\left(x;P\right)=1/(d+1)$.

###  A.11. Proof of Theorem 10

We have

(38) $\displaystyle\sup_{x\in\mathbb{S}^{d-1}}$
$\displaystyle\left|ahD(x;P_{n})-ahD(x;P)\right|=\sup_{x\in\mathbb{S}^{d-1}}\left|\inf_{H\in\mathcal{H}_{0}\colon
x\in H}P_{n}(H)-\inf_{H\in\mathcal{H}_{0}\colon x\in H}P(H)\right|$
$\displaystyle\leq\sup_{x\in\mathbb{S}^{d-1}}\sup_{H\in\mathcal{H}_{0}\colon
x\in
H}\left|P_{n}(H)-P(H)\right|\leq\sup_{H\in\mathcal{H}_{0}}\left|P_{n}(H)-P(H)\right|.$

The final expression vanishes as $n\to\infty$ by, e.g., Nagy et al., (2016,
Theorem A.3).

###  A.12. Proof of Theorem 11

We apply the proof of implication (ComD) $\Rightarrow$ (ComR) from Dyckerhoff,
(2017, Theorem 4.5). In that paper, only depths in $\mathbb{R}^{d}$ are
considered. Nevertheless, the fact that one works in $\mathbb{R}^{d}$ is used
only in several arguments in the proofs, and many of the derivations work
precisely in the same way for measures in $\mathbb{S}^{d-1}$.

For starters, in Dyckerhoff, (2017, page 4) it is assumed that the function
$D$ in $\mathbb{R}^{d}$ must be a depth in the sense of Definition 2.1 in
Dyckerhoff, (2017). That requires that the upper level sets
$D_{\alpha}(P)=\left\\{x\in\mathbb{R}^{d}\colon D(x;P)\geq\alpha\right\\}$ are
all  (R1) affine equivariant (for $\alpha\geq 0$),  (R2) bounded (for
$\alpha>0$),  (R3) closed (for $\alpha>0$), and  (R4) star-shaped (for all
$\alpha\geq 0$).  The requirement (R1) of affine equivariance is, however, not
used anywhere in the proof of Theorem 4.5 in Dyckerhoff, (2017). The
(spherical counterparts of the) additional conditions R2–R4 are all satisfied
for ahD, because  • $ahD_{\alpha}(P)\subseteq\mathbb{S}^{d-1}$for all
$\alpha\geq 0$ and $\mathbb{S}^{d-1}$ is itself bounded, giving R2,  • each
$ahD_{\alpha}(P)$ is closed by Theorem 3, which verifies R3, and  •
(spherical) star-shapedness of $ahD_{\alpha}(P)$ in R4 follows immediately
from the (spherical) convexity of $ahD_{\alpha}(P)$ as proved in Theorem 4.
Thus, the advances proved in Dyckerhoff, (2017) can be used in our setup.

Mimicking the proof of Theorem 4.5 in Dyckerhoff, (2017) for a depth $D$ in
$\mathbb{R}^{d}$, we see that the core of the proof actually lies in proving
• the continuity of the function $\alpha\mapsto D_{\alpha}(P)$ in the
Hausdorff metric (23), and  • the bound

(39) $D_{\beta+\gamma}(P)\subseteq D_{\beta}(P_{n})\subseteq
D_{\beta-\gamma}(P)\quad\mbox{for all $n\geq N$ and $\beta\geq\alpha_{1}$},$

where $\alpha_{1}$ is the lower endpoint of the interval
$A\subset(0,\max_{x\in\mathbb{R}^{d}}D(x;P))$, $\gamma>0$ is a small enough
constant, and $N$ is a large enough integer.

The continuity of the mapping $\alpha\mapsto D_{\alpha}(P)$ is obtained in
Dyckerhoff, (2017, Theorem 3.2). There, it is asserted that strict
monotonicity of the depth $D$ for
$P\in\mathcal{P}\left({\mathbb{R}^{d}}\right)$ at all
$\alpha\in(0,\max_{x\in\mathbb{R}^{d}}D(x;P))$ is enough to have continuity of
$\alpha\mapsto D_{\alpha}(P)$ in
$\alpha\in(0,\max_{x\in\mathbb{R}^{d}}D(x;P)]$. The proof of Theorem 3.2 in
Dyckerhoff, (2017) does not use the fact that we work in $\mathbb{R}^{d}$; it
works equally well in $\mathbb{S}^{d-1}$. In our setup, however, we assume
strict monotonicity in (25) only for
$\alpha\in(\min_{x\in\mathbb{S}^{d-1}}ahD(x;P),\max_{x\in\mathbb{S}^{d-1}}ahD(x;P))$,
which by the same argument gives continuity of $\alpha\mapsto ahD_{\alpha}(P)$
in the Hausdorff distance (23) for
$\alpha\in(\min_{x\in\mathbb{S}^{d-1}}ahD(x;P),\max_{x\in\mathbb{S}^{d-1}}ahD(x;P)]$.

As the second ingredient, we need to prove (39) for $D=ahD$. That is
implication $(ii)\Rightarrow(iii)$ in Dyckerhoff, (2017, Theorem 4.4). As
before, the proof of this implication does not use the fact that the depth $D$
is defined in $\mathbb{R}^{d}$ and also works for $ahD$ in $\mathbb{S}^{d-1}$.
In fact, the proof can be simplified by dropping the argument of a bounded set
$M$ since, in our setup, we could take $M=\mathbb{S}^{d-1}$ directly. In view
of our restriction to $\alpha>\min_{x\in\mathbb{S}^{d-1}}ahD(x;P)$ in proving
continuity of $ahD_{\alpha}(P)$, we obtain (39) for all
$\beta\geq\alpha_{1}>\min_{x\in\mathbb{S}^{d-1}}ahD(x;P)$.

Putting the two ingredients together, in the proof of implication (ComD)
$\Rightarrow$ (ComR) from Dyckerhoff, (2017, Theorem 4.5) we obtain (27) as
needed.

Observe that in Dyckerhoff, (2017), two additional conditions are assumed for
the depth $D$ throughout the paper:

  * •

The range condition (RC) from page 9 of Dyckerhoff, (2017).

  * •

The “general assumption” stating that for each
$\alpha<\max_{x\in\mathbb{S}^{d-1}}ahD(x;P)$ must the set $ahD_{\alpha}(P)$
have non-empty interior.

None of these is necessary for the proof of (27).

###  A.13. Proof of Theorem 12

The condition of existence of a dominant hemisphere immediately implies that
$\mathbb{S}^{d-1}\setminus H_{u}$ is an open hemisphere of minimum depth as in
Theorem 6. The smoothness of $P$ and the continuity of ahD in $x$ from Theorem
3 then assert that the whole closed hemisphere $\mathbb{S}^{d-1}\cap H_{-u}$
is of minimum ahD. To prove the strict monotonicity of ahD, we pick any point
$x\in\mathbb{S}^{d-1}$ from the ahD-median set of $P$. Because of the
smoothness of $P$, we can argue as in Laketa and Nagy, (2022, Lemma 10) to
show that $ahD_{\alpha}(P)$ must have non-empty (spherical) interior for all
$\alpha\leq\max_{y\in\mathbb{S}^{d-1}}ahD(y;P)$. Further, the dominant
hemisphere condition says that for any $H,H^{\prime}\in\mathcal{H}_{0}$ such
that $H\cap H_{u}\subset H^{\prime}\cap H_{u}$ we have

$P(H^{\prime})=P(H^{\prime}\cap H)+P(H^{\prime}\setminus H)>P(H^{\prime}\cap
H)+P(H\setminus H^{\prime})=P(H),$

since $H^{\prime}\cap H\subset H_{u}$. This allows us to use the arguments as
in the linear case and hD in parts (iii) and (v) of the proof of Laketa and
Nagy, (2022, Theorem 9), which together give (25).

###  A.14. Proof of Theorem 13

Our first claim can be proved analogously as for Theorem 10. The only
exception is that in the final step, the last expression in (38) vanishes as
$n\to\infty$ almost surely, because the set of closed halfspaces $\mathcal{H}$
and its subset $\mathcal{H}_{0}$ are both Glivenko-Cantelli classes of sets
(van der Vaart and Wellner,, 1996, Section 2.4 and Problem 14 in Section 2.6).

For the second claim, part (i) gives that (26) is valid for
$\mathsf{P}$-almost all $\omega\in\Omega$. Thus, we can separately apply
Theorem 11 for each $\omega\in\Omega$ and obtain the result.

The last part of our theorem follows by direct application of part (iii) of
Theorem 9 and the theorem of Varadarajan (see, e.g., Dudley,, 2002, Theorem
11.4.1).

### Acknowledgment

The work of Stanislav Nagy was supported by Czech Science Foundation (EXPRO
project n. 19-28231X).

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