# A note on faces of convex sets

Stephan Weis

(Date: 31 March, 2024)

###### Abstract.

At the core of convex analysis, we find the concept of a face of a convex set,
which was systematically studied by R. T. Rockafellar. With the exception of
the one-point faces known as extreme points, faces received little attention
in the theory of infinite-dimensional convexity, perhaps due to their lack of
relative interior points. Circumventing this peculiarity, M. E. Shirokov and
the present author explored faces generated by points. These faces possess
relative interior points and they can serve as building blocks for general
faces. Here we deepen this approach, taking into account ideas by E. M. Alfsen
and L. E. Dubins. We also indicate possible application in probability theory.

###### Key words and phrases:

Convex set, face, face generated by a point, relative interior, Radon-Nikodym
derivative, convex core

###### 2020 Mathematics Subject Classification:

52A05, 46E27

Dedicated to R. Tyrrell Rockafellar on His Ninetieth Birthday

##  1\. Introduction

The face of a convex set is a key concept in convex analysis, and R. T.
Rockafellar [10] studies it systematically in his work. Extreme points, which
are defined as points that can not be written as proper convex combinations,
are very useful in infinite-dimensional convexity [1, 4, 5, 14]. They are in a
one-to-one correspondence with the one-point faces. General faces (which are
defined as convex subsets containing the endpoints of each open segment they
intersect) are, however, less important in infinite-dimensional convexity. We
are aware of exceptions, notably in the works of E. M. Alfsen and F. W. Shultz
on state spaces of operator algebras [2], where norm-closed faces play an
essential role.

Perhaps, a lack of relative interior points is the reason why faces received
little attention in the theory of infinite-dimensional convexity. The relative
interior of a finite-dimensional convex set $K$ is the interior with respect
to the affine hull $\operatorname{aff}(K)$ of $K$, and it has the key property
that its closure contains $K$ [11, p. 75]. Here, we define a relative interior
point of a convex set $K$ as a point that lies in an open segment in $K\cap g$
for every line $g\subset\operatorname{aff}(K)$ incident with $x$. This
definition is equivalent (by Thm. 3.3) to that of an “internal point”, a point
$x\in K$ with the property that for all $y\in K$ there is $\epsilon>0$ such
that $x+\epsilon(x-y)\in K$. The term “internal point” appeared in the works
of L. E. Dubins [7] and N. Bourbaki [5, TVS II.26] on convex geometry and
functional analysis. The related term “algebraic interior” was used by A.
Barvinok [4]. Although a nonempty convex set may have no relative interior
points, it certainly does when its dimension is finite. Then, the two
definitions of relative interior points mentioned above, are equivalent.

A key to the geometry of a convex set $K$ is the face $F_{K}(x)$ generated by
a point $x\in K$, which is the smallest face of $K$ containing $x$. M. E.
Shirokov and the present author proved that $x$ lies in the relative interior
$\,\operatorname{ri}(F_{K}(x))$ of $F_{K}(x)$, and explored consequences
thereof [13].

In this paper, we note that $x\in\operatorname{ri}(F_{K}(x))$ follows directly
from the definition of an “internal point” above, and from E. M. Alfsen’s
formula [1, p. 121]

$F_{K}(x)=\left\\{y\in K\mid\exists\epsilon>0\colon x+\epsilon(x-y)\in
K\right\\}\,\text{.}$ (1.1)

We dwell on this observation in Sec. 3 as it greatly simplifies the prior
proof, which employs the Kuratowski-Zorn lemma [13, Thm. 2.3].

Using the faces $F_{K}(x)$ as building blocks of general faces, we generalize
a number of results familiar from finite dimensions. Sec. 4 reviews prior
results on that matter [13, Sec. 2]. Sec. 5 is dedicated to generalizing
results from Sec. 6 “Relative Interiors of Convex Sets” and Sec. 18 “Extreme
Points and Faces of Convex Sets” in R. T. Rockafellar’s monograph [10] into
the infinite-dimensional setting. Sec. 6 and Sec. 7 is a (nonexhaustive)
review of L. E. Dubins’s paper [7] on infinite-dimensional convex geometry,
which employs a different notion of a “face”.

Indicating potential applications in probability theory, we improve
observations by L. E. Dubins [7], and by M. E. Shirokov and this author [13]
on the faces of the convex set of probability measures (Sec. 8). We apply our
methods to I. Csiszár and F. Matúš’s [6] concept of the convex core (Sec. 9).
We highlight examples of faces lacking relative interior points and of faces
failing to be closed in the distance in variation (Sec. 10).

##  2\. Definitions of faces and relative interiors

Let $V$ be a real vector space. The _open segment_ resp.  _closed segment_
with endpoints $x,y\in V$ is the set

$\displaystyle(x,y)$ $\displaystyle=\\{(1-\lambda)x+\lambda
y\colon\lambda\in(0,1)\\}$ $\displaystyle{}\text{resp.}\quad[x,y]$
$\displaystyle=\\{(1-\lambda)x+\lambda y\colon\lambda\in[0,1]\\}\,\text{.}$

Here, $(0,1)=\\{x\in\mathbb{R}\colon 0<x<1\\}$ and
$[0,1]=\\{x\in\mathbb{R}\colon 0\leq x\leq 1\\}$. If $x\neq y$, then $(x,y)$
resp. $[x,y]$ is a _proper open segment_ resp. a _proper closed segment_. The
segment $(x,x)=[x,x]=\\{x\\}$ is an _improper segment_ for all $x\in V$.

Throughout this paper, $K$ is a convex subset of $V$ unless stated otherwise.
An _extreme set_ [9] of $K$ is a subset $E$ of $K$ such that, whenever an open
segment with endpoints $x,y\in K$ intersects $E$, then the closed segment
$[x,y]$ is included in $E$. A _face_ of $K$ is a convex extreme set of $K$. A
point $x\in K$ is an _extreme point_ of $K$ if $\\{x\\}$ is a face of $K$.
Clearly, any union or intersection of extreme sets of $K$ is an extreme set of
$K$. Since any intersection of convex sets is a convex set, any intersection
of faces of $K$ is a face of $K$. Hence, the intersection of all faces
containing a given point $x\in K$ is a face of $K$, which we call the _face of
$K$ generated by $x$_ and which we denote by $F_{K}(x)$.

We say a point $x\in K$ is a _relative interior point_ of $K$ if, whenever $x$
lies on a line $g$ included in the affine hull of $K$, then $x$ lies in a
proper open segment included in $g\cap K$. The _relative interior_ of $K$ is
the set of all relative interior points of $K$, which we denote by
$\operatorname{ri}(K)$. The set $K$ is _relative open_ if every $x\in K$ is a
relative interior point of $K$, or equivalently, if $K=\operatorname{ri}(K)$.

##  3\. Faces generated by points have relative interior points

We give an elementary proof of the assertion that every point $x\in K$ lies in
$\operatorname{ri}(F_{K}(x))$. We use E. M. Alfsen’s formula (1.1) and a
characterization of relative interior points without reference to the affine
hull.

Let $x\in K$ and let

$S_{K}(x)=\left\\{y\in K\mid\exists\epsilon>0\colon x+\epsilon(x-y)\in
K\right\\}\,\text{.}$

a)b)

Figure 1. Configurations (in the plane) for Thm. 3.1. The set $S_{K}(x)$ is a)
an extreme set and b) convex.

######  Theorem 3.1 (Alfsen).

We have $F_{K}(x)=S_{K}(x)$.

###### Proof.

If $y\in S_{K}(x)$, then there is $\epsilon>0$ such that $x$ lies in the open
segment with endpoints $x+\epsilon(x-y)$ and $y$, hence $y\in F_{K}(x)$, as
$F_{K}(x)$ is an extreme set containing $x$.

We finish the proof by showing that $S_{K}(x)$ is a face of $K$. Let $y$ be a
point in the open segment $(a,b)$ with endpoints $a,b\in K$. More precisely,
let $y=(1-\eta)a+\eta b$ for some $\eta\in(0,1)$.

We show that $S_{K}(x)$ is an extreme set. Let $\epsilon>0$ such that
$y^{\prime}=x+\epsilon(x-y)$ lies in $K$, and let
$\epsilon_{a}=\epsilon(1-\eta)/(1+\epsilon\eta)$ and
$\xi_{a}=\epsilon\eta/(1+\epsilon\eta)$. Then

$a^{\prime}=x+\epsilon_{a}(x-a)=(1-\xi_{a})y^{\prime}+\xi_{a}b$

shows $a\in S_{K}(x)$. The scalars $\epsilon_{a}$ and $\xi_{a}$ above, and
those in the next paragraph below, are obtained from Menelaus’s theorem [3]
applied to the configurations in Fig. 1. Replacing $\eta$ with $1-\eta$, we
obtain $b\in S_{K}(x)$ by the same approach.

To show that $S_{K}(x)$ is convex, we assume $a,b\in S_{K}(x)$. Let
$\epsilon_{a},\epsilon_{b}>0$ such that $a^{\prime}=x+\epsilon_{a}(x-a)$ and
$b^{\prime}=x+\epsilon_{b}(x-b)$ lie in $K$. Let

$\textstyle\epsilon=\frac{\epsilon_{a}\epsilon_{b}}{(1-\eta)\epsilon_{b}+\eta\epsilon_{a}}\quad\text{and}\quad\xi=\frac{\eta\epsilon_{a}}{(1-\eta)\epsilon_{b}+\eta\epsilon_{a}}\,\text{.}$

Then

$y^{\prime}=x+\epsilon(x-y)=(1-\xi)a^{\prime}+\xi b^{\prime}$

shows $y\in S_{K}(x)$. ∎

The union in Coro. 3.2 extends over all closed segments in $K$ for which $x$
lies on the corresponding open segment, the improper segment $\\{x\\}$
included.

######  Corollary 3.2.

For all $x\in K$ we have $F_{K}(x)=\bigcup_{y,z\in K,x\in(y,z)}[y,z]$.

###### Proof.

If $a\in S_{K}(x)$, then there is $\epsilon>0$ such that $b=x+\epsilon(x-a)\in
K$. Then $x=\frac{1}{1+\epsilon}(\epsilon a+b)\in(a,b)$. Conversely, if
$x=(1-\eta)a+\eta b$ for some $\eta\in(0,1)$ and $a,b\in K$, then
$b=x+\frac{1-\eta}{\eta}(x-a)$ shows $a\in S_{K}(x)$. ∎

Thm. 3.3 matches [10, Thm. 6.4].

######  Theorem 3.3.

A point $x\in K$ is a relative interior point of $K$ if and only if for all
$y\in K$ there is $\epsilon>0$ such that $x+\epsilon(x-y)\in K$.

###### Proof.

Let $x\in\operatorname{ri}(K)$ be a relative interior point of $K$ and let
$y\in K$. The intersection of the line $g$ spanned by $x$ and $y$ with $K$
contains a proper open segment about $x$ by the definition of relative
interior points. This implies $x+\epsilon(x-y)\in K$ for some $\epsilon>0$.

Conversely, let $g\subset\operatorname{aff}(K)$ be a line incident with $x$
and let $y\in g$ be a point on $g$ different from $x$. There are $y_{i}\in K$
and $\alpha_{i}\in\mathbb{R}$ for $i=1,\dots,n$, such that

$\textstyle
y=x+\sum_{i}\alpha_{i}y_{i}\quad\text{and}\quad\sum_{i}\alpha_{i}=0\,\text{.}$

By assumption, there is $\epsilon_{i}>0$ such that

$y_{i}^{\prime}=x+\epsilon_{i}(x-y_{i})\in K\quad i=1,\dots,n\,\text{.}$

By the convexity of $K$ we are allowed to choose $\epsilon_{i}\in(0,1]$ to be
at most one.

We complete the proof by showing that the two points $x\pm\epsilon(x-y)$ lie
in $K$ when $\epsilon=\min_{i}\epsilon_{i}/\|\alpha\|_{1}$. Here
$\|\alpha\|_{1}=\sum_{i=1}^{n}|\alpha_{i}|$ denotes the $1$-norm of the vector
$\alpha=(\alpha_{1},\dots,\alpha_{n})$. We write

$\displaystyle\textstyle x\pm\epsilon(x-y)$
$\displaystyle\textstyle=x\mp\epsilon\sum_{i}\alpha_{i}y_{i}=x\pm\epsilon\sum_{i}\alpha_{i}(x-y_{i})$
$\displaystyle\textstyle=\sum_{i}\frac{|\alpha_{i}|}{\|\alpha\|_{1}}\underbrace{\left(x\pm\operatorname{sgn}(\alpha_{i})\epsilon\|\alpha\|_{1}(x-y_{i})\right)}_{=z_{i}}$

as a convex combinations of points $z_{i}$, $i=1,\dots,n$. If
$\pm\operatorname{sgn}(\alpha)=+1$, then
$\epsilon\|\alpha\|_{1}\leq\epsilon_{i}$ implies that
$z_{i}\in[y_{i}^{\prime},x]\subset K$. Otherwise, if
$\pm\operatorname{sgn}(\alpha)=-1$, then $\epsilon\|\alpha\|_{1}\leq 1$
implies $z_{i}\in[x,y_{i}]\subset K$. ∎

We are ready for the main result of this section.

######  Corollary 3.4.

For all $x\in K$ we have $x\in\operatorname{ri}(F_{K}(x))$.

###### Proof.

This follows immediately from Thm. 3.1 and Thm. 3.3. ∎

Let $x\in K$ and let

$A_{K}(x)=\left\\{y\in V\mid\exists\epsilon>0\colon x\pm\epsilon(x-y)\in
K\right\\}\,\text{.}$

######  Lemma 3.5.

If $x\in\operatorname{ri}(K)$, then $\operatorname{aff}(K)=A_{K}(x)$.

###### Proof.

This follows directly from the definition of relative interior points. ∎

######  Corollary 3.6.

For all $x\in K$ we have $\,\operatorname{aff}\left(F_{K}(x)\right)=A_{K}(x)$.

###### Proof.

We have $\operatorname{aff}\left(F_{K}(x)\right)=A_{F_{K}(x)}(x)$ by Lemma 3.5
as $x\in\operatorname{ri}(F_{K}(x))$ by Coro. 3.4. As $F_{K}(x)\subset K$, we
get $A_{F_{K}(x)}(x)\subset A_{K}(x)$.

The converse is easier. If $y\in A_{K}(x)$, then there is $\epsilon>0$ such
that $z_{\pm}=x\pm\epsilon(x-y)\in K$. Moreover, $z_{\pm}\in F_{K}(x)$ holds
as $F_{K}(x)$ is an extreme set containing $x$. Hence, $y$ is an affine
combination of points in $F_{K}(x)$,

$\textstyle
y=\frac{1}{2\epsilon}\left(\epsilon-1\right)z_{+}+\frac{1}{2\epsilon}\left(\epsilon+1\right)z_{-}\,\text{.}$

∎

##  4\. A review on the face generated by a point

In addition to the simplified proofs of our prior work [13, Sec. 2] in Sec. 3
above, we review some more aspects of faces generated by points in a slightly
extended form.

Lemma 4.1.1 matches [10, Thm. 18.1].

######  Lemma 4.1.

Let $C\subset K$ be a convex subset of $K$, let $E\subset K$ be an extreme set
of $K$, let $F\subset K$ be a face of $K$, and let $x\in K$ be a point in $K$.
Then

  1. 1)

$\operatorname{ri}(C)\cap E\neq\emptyset\implies C\subset E$,

  2. 2)

$x\in F\iff F_{K}(x)\subset F$,

  3. 3)

$x\in\operatorname{ri}(F)\implies F=F_{K}(x)$.

###### Proof.

See Lemma 2.1 in [13]. ∎

Lemma 4.2 matches part of [10, Thm. 6.2].

######  Lemma 4.2.

The complement $K\setminus\operatorname{ri}(K)$ of the relative interior
$\operatorname{ri}(K)$ is an extreme set of $K$ and $\operatorname{ri}(K)$ is
a convex set.

###### Proof.

See Lemma 2.2 in [13]. ∎

The remainder of this review relies on the assertion of
$x\in\operatorname{ri}(F_{K}(x))$ for all $x\in K$, which is proved in [13,
Thm. 2.3] and in Coro. 3.4 above.

######  Corollary 4.3.

The following assertions are equivalent. A subset $C\subset K$

  1. 1)

is an extreme set of $K$,

  2. 2)

contains the face $F_{K}(x)$ of $K$ generated by any point $x$ in $C$,

  3. 3)

is the union of the faces $F_{K}(x)$ of $K$ generated by the points $x$ in
$C$,

  4. 4)

is a union of the faces $F_{K}(x)$ of $K$ generated by points $x$ in $K$,

  5. 5)

is a union of faces of $K$,

  6. 6)

is a union of extreme sets of $K$.

###### Proof.

See Coro. 2.5 in [13]. We provide a complete proof for easy reference.

1) $\Rightarrow$ 2) follows from Coro. 3.4 and Lemma 4.1.1. 2) $\Rightarrow$
3) is true as we have $x\in F_{K}(x)$ for all $x\in K$. The implications 3)
$\Rightarrow$ 4), 4) $\Rightarrow$ 5), and 5) $\Rightarrow$ 6) are clear. 6)
$\Rightarrow$ 1) follows because a union of extreme sets is an extreme set. ∎

The following Coro. 4.4, and Thm. 5.9 below, match [10, Thm. 18.2].

######  Corollary 4.4.

The families (4.1) and (4.2) of subsets of $K$ are equal.

$\displaystyle\left\\{\operatorname{ri}(F_{K}(x))\colon x\in K\right\\}$ (4.1)
$\displaystyle\left\\{\operatorname{ri}(F)\colon\text{$F$ is a face of
$K$}\right\\}\setminus\\{\emptyset\\}$ (4.2)

Each of the families is a partition of the convex set $K$, that is to say, the
sets in each family are disjoint and their union is $K$.

###### Proof.

See Coro. 2.6 in [13]. We provide a complete proof for easy reference.

The union over the family (4.1) covers $K$ as
$x\in\operatorname{ri}(F_{K}(x))$ by Coro. 3.4, and so does the union over the
larger family (4.2). Consider two members of the family (4.2) that intersect
in a point $x\in\operatorname{ri}(F)\cap\operatorname{ri}(G)$, where $F,G$ are
faces of $K$. Lemma 4.1.3 shows $F=F_{K}(x)$ and Lemma 4.1.2 shows
$F_{K}(x)\subset G$, hence $F\subset G$. The converse inclusion follows
analogously. ∎

######  Remark 4.5 (Building blocks of faces).

  1. 1)

A face of $K$ that cannot be generated by a point of $K$ must have an empty
relative interior, as the families (4.1) and (4.2) in Coro. 4.4 cover $K$.
Examples of convex sets with empty relative interiors are given in [4, Sec.
III.1.6], [13, Sec. 2], and below in Sec. 8.

  2. 2)

Every extreme set resp. face of $K$ is a union resp. convex union of members
of the family (4.1) by Coro. 3.4 and Lemma 4.1.1. The converse is false. For
example, the open unit interval $(0,1)$ is neither an extreme set nor a face
of the closed unit interval $[0,1]$.

  3. 3)

A modified converse to part 2) is true if our notion of a face is replaced
with that by L. E. Dubins, which we characterize in Thm. 7.1.

######  Corollary 4.6.

Let $F$ be a face of $K$ and $x\in K$. The following statements are
equivalent.

  1. 1)

We have $x\in\operatorname{ri}(F)$.

  2. 2)

We have $F=F_{K}(x)$.

  3. 3)

We have $\operatorname{ri}(F)=\operatorname{ri}(F_{K}(x))$.

###### Proof.

1) $\Rightarrow$ 2) is Lemma 4.1.3. 2) $\Rightarrow$ 3) is clear. 3)
$\Rightarrow$ 1) is implied by Coro. 3.4. (See also [13, Coro. 2.7].) ∎

Prop. 4.7 complements [7, Thm. 4.3] but is not equivalent to it, as different
concepts of a “face” are in use (see Sec. 7).

######  Proposition 4.7.

Let $K,L\subset V$ be two convex sets and let $x\in K\cap L$. Then

  1. 1)

$F_{K\cap L}(x)$

$=$ $F_{K}(x)\cap F_{L}(x)$,

  2. 2)

$\operatorname{ri}\big{(}F_{K\cap L}(x)\big{)}$

$=$
$\operatorname{ri}\big{(}F_{K}(x)\big{)}\cap\operatorname{ri}\big{(}F_{L}(x)\big{)}$,

  3. 3)

$\operatorname{aff}\big{(}F_{K\cap L}(x)\big{)}$

$=$
$\operatorname{aff}\big{(}F_{K}(x)\big{)}\cap\operatorname{aff}\big{(}F_{L}(x)\big{)}$.

If $F$ is a face of $K\cap L$ with nonempty relative interior
$\operatorname{ri}(F)\neq\emptyset$, then the equations 1)–3) hold for all
$x\in\operatorname{ri}(F)$ when $F_{K\cap L}(x)$ is placed with $F$. A
sufficient condition for $\operatorname{ri}(F)\neq\emptyset$ is that the face
$F$ have finite dimension.

###### Proof.

The assertions 1)–3) are proved in [13, Prop. 2.13]. Let
$x\in\operatorname{ri}(F)$. By Lemma 4.1.3 we have $F=F_{K\cap L}(x)$, from
which the second claim follows. Nonempty finite-dimensional convex sets have
relative interior points by Thm. 6.2 in [10]. ∎

Prop. 4.7 does not generalize to infinite intersections.

######  Example 4.8 (Infinite intersections).

For all $\epsilon>0$ we consider the open segment
$C_{\epsilon}=(-\epsilon,1+\epsilon)$ and the closed segment
$D_{\epsilon}=[-\epsilon,1+\epsilon]$. The intersection
$\bigcap_{\epsilon>0}C_{\epsilon}=\bigcap_{\epsilon>0}D_{\epsilon}$ is the
unit interval $[0,1]$. However, the extreme points $0$ and $1$ of $[0,1]$
cannot be written as intersections of faces of $C_{\epsilon}$ nor
$D_{\epsilon}$.

Is the assumption of $\operatorname{ri}(F)\neq\emptyset$ in Prop. 4.7
necessary?

######  Open Problem 4.9.

Are there convex sets $K$ and $L$ and a face $F$ of the intersection $K\cap
L$, such that $F$ cannot be written as the intersection of a face of $K$ and a
face of $L$?

##  5\. Novel results: Relative interior points

This section is inspired by properties of relative interiors of convex sets in
finite dimensions [10].

######  Lemma 5.1.

Let $x,y\in K$. Then $y\in\operatorname{ri}(F_{K}(x))$ if and only if $y\in
F_{K}(x)$ and $x\in F_{K}(y)$.

###### Proof.

If $y\in\operatorname{ri}(F_{K}(x))$ then $F_{K}(x)=F_{K}(y)$ by Lemma 4.1.3,
and hence $y\in F_{K}(x)$ and $x\in F_{K}(y)$. Conversely, if $x\in F_{K}(y)$
then Lemma 4.1.2 gives $F_{K}(x)\subset F_{K}(y)$. Swapping $x$ and $y$ gives
$F_{K}(x)=F_{K}(y)$. Now Coro. 3.4 shows $y\in\operatorname{ri}(F_{K}(x))$,
completing the proof. ∎

The union in Lemma 5.2 extends over all open segments in $K$ that contain the
point $x$, including the improper segment $\\{x\\}$.

######  Lemma 5.2.

For all $x\in K$ we have $\operatorname{ri}(F_{K}(x))=\bigcup_{y,z\in
K,x\in(y,z)}(y,z)$.

###### Proof.

The point $x$ belonging to $\bigcup_{y,z\in K,x\in(y,z)}(y,z)$ is consistent
with $x$ belonging to $\operatorname{ri}(F_{K}(x))$, which holds by Coro. 3.4.
Let $a\in K$ and $a\neq x$. Then

$\begin{array}[]{rcl}a\in\operatorname{ri}(F_{K}(x))&\stackrel{{\scriptstyle\rm{Lemma\leavevmode\nobreak\
\ref{lem:char1-FxFy}}}}{{\iff}}&\mbox{$a\in F_{K}(x)$ and $x\in F_{K}(a)$}\\\
&\stackrel{{\scriptstyle\rm{Coro.\leavevmode\nobreak\
\ref{cor:facex}}}}{{\iff}}&\exists y,z\in K:\mbox{$x\in(a,y)$ and
$a\in(x,z)$}\\\ &\stackrel{{\scriptstyle a\neq x}}{{\iff}}&\exists y,z\in
K:x,a\in(y,z)\,\text{,}\end{array}$

which proves the claim. ∎

######  Corollary 5.3.

Let $x\in K$. The following statements are equivalent.

  1. 1)

The point $x$ lies in $\operatorname{ri}(K)$.

  2. 2)

For all $y\in K$ there is $\epsilon>0$ such that $x+\epsilon(x-y)\in K$.

  3. 3)

We have $K=F_{K}(x)$.

  4. 4)

We have $K=\bigcup_{y,z\in K,x\in(y,z)}[y,z]$.

  5. 5)

We have $\operatorname{ri}(K)=\operatorname{ri}(F_{K}(x))$.

  6. 6)

We have $\operatorname{ri}(K)=\bigcup_{y,z\in K,x\in(y,z)}(y,z)$.

###### Proof.

The statements 1), 3), and 5) are equivalent by Coro. 4.6. The equivalences 1)
$\Leftrightarrow$ 2), 3) $\Leftrightarrow$ 4), and 5) $\Leftrightarrow$ 6)
follow from Thm. 3.3, Coro. 3.2, and Lemma 5.2, respectively. ∎

Coro. 5.4 matches one of the statements of [10, Thm. 6.2].

######  Corollary 5.4.

If $\,\operatorname{ri}(K)\neq\emptyset$, then
$\operatorname{aff}(\operatorname{ri}(K))=\operatorname{aff}(K)$ holds.

###### Proof.

Let $x\in\operatorname{ri}(K)$ and $y\in\operatorname{aff}(K)$. Lemma 3.5
shows that there is $\epsilon>0$ such that $x\pm\epsilon(x-y)\in K$. Hence
$y_{\pm}=x\pm\frac{\epsilon}{2}(x-y)\in\operatorname{ri}(K)$ by Coro. 5.3.6,
which implies

$\textstyle
y=\left(\frac{1}{2}-\frac{1}{\epsilon}\right)y_{+}+\left(\frac{1}{2}+\frac{1}{\epsilon}\right)y_{-}\in\operatorname{aff}(\operatorname{ri}(K))\,\text{.}$

The opposite inclusion is obvious. ∎

######  Theorem 5.5.

The set $\operatorname{ri}(K)$ is a relative open convex set.

###### Proof.

The convexity of $\operatorname{ri}(K)$ is provided by Lemma 4.2. It remains
to prove $\operatorname{ri}(K)\subset\operatorname{ri}(\operatorname{ri}(K))$,
as this implies that $\operatorname{ri}(K)$ is relative open. Let
$x\in\operatorname{ri}(K)$. Thm. 3.3 shows that instead of
$x\in\operatorname{ri}(\operatorname{ri}(K))$ we can equivalently prove that
for all $y\in\operatorname{ri}(K)$ there is $\epsilon>0$ such that
$x+\epsilon(x-y)$ lies in $\operatorname{ri}(K)$. Because of
$x,y\in\operatorname{ri}(K)$, Coro. 5.3.6 provides $a,b\in K$ such that $x$
and $y$ lie in the open segment $(a,b)$, say $x=(1-\lambda)a+\lambda b$ and
$y=(1-\mu)a+\mu b$, where $0<\lambda<\mu<1$, and such that the open segment
$(a,b)$ is included in $\operatorname{ri}(K)$. Hence, the point $(a+x)/2$ lies
in $\operatorname{ri}(K)$, and

$\textstyle\frac{1}{2}(a+x)=x+\frac{\lambda}{2(\mu-\lambda)}(x-y)\,\text{.}$

∎

Coro. 5.6 matches the second part of [10, Thm. 6.5].

######  Corollary 5.6.

If $K,L\subset V$ are two convex sets and
$\operatorname{ri}(K)\cap\operatorname{ri}(L)\neq\emptyset$, then

$\operatorname{ri}(K\cap L)=\operatorname{ri}(K)\cap\operatorname{ri}(L)$

is a relative open convex set. The intersection of two relative open convex
sets is relative open.

###### Proof.

If $x$ lies in $\operatorname{ri}(K)\cap\operatorname{ri}(L)$, then
$K=F_{K}(x)$ and $L=F_{L}(x)$ follows from Lemma 4.1.3 and Prop. 4.7 gives
$\operatorname{ri}(K\cap L)=\operatorname{ri}(K)\cap\operatorname{ri}(L)$.
Thm. 5.5 proves that $\operatorname{ri}(K\cap L)$ is a relative open convex
set. The second assertion is a special case of the first one. ∎

The second assertion of Coro. 5.6 follows also from Coro. 5.10 and Coro. 6.3.
Thm. 5.7 matches the first statement of [10, Thm. 6.6].

######  Theorem 5.7.

If $\operatorname{ri}(K)\neq\emptyset$ and if $\alpha:V\to W$ is an affine map
to a real vector space $W$, then
$\operatorname{ri}(\alpha(K))=\alpha(\operatorname{ri}(K))$.

###### Proof.

First, we use Thm. 3.3 to show that $\alpha(x)\in\operatorname{ri}(\alpha(K))$
if $x\in\operatorname{ri}(K)$. It suffices to show that for $y\in\alpha(K)$
there is $\epsilon>0$ such that $\alpha(x)+\epsilon(\alpha(x)-y)$ lies in
$\alpha(K)$. Taking $y^{\prime}\in\alpha|_{K}^{-1}(y)$, there exists
$\epsilon>0$ such that $x+\epsilon(x-y^{\prime})$ lies in $K$ as
$x\in\operatorname{ri}(K)$. The claim follows by applying $\alpha$ to
$x+\epsilon(x-y^{\prime})$.

Coro. 5.3.6 implies

$\displaystyle\alpha(\operatorname{ri}(K))$
$\displaystyle=\textstyle\alpha\left(\bigcup_{y,z\in K,x\in(y,z)}(y,z)\right)$
$\displaystyle=\textstyle\bigcup_{y,z\in K,x\in(y,z)}(\alpha(y),\alpha(z))$
$\displaystyle=\textstyle\bigcup_{y^{\prime},z^{\prime}\in\alpha(K),\alpha(x)\in(y^{\prime},z^{\prime})}(y^{\prime},z^{\prime})\,\text{.}$

As $\alpha(x)\in\operatorname{ri}(\alpha(K))$, the right-hand side of this
equation equals $\operatorname{ri}(\alpha(K))$, again by Coro. 5.3.6. ∎

######  Corollary 5.8.

If $C\subset K$ is a relative open convex subset, then we have
$C\subset\operatorname{ri}(F_{K}(x))$ for all $x\in C$.

###### Proof.

Let $x,y\in C$. Coro. 5.3.6 shows that there are $a,b\in C$ such that
$x,y\in(a,b)$. Then Lemma 5.2 shows $y\in\operatorname{ri}(F_{K}(x))$. ∎

The following Thm. 5.9 generalizes [10, Thm. 18.2] from finite dimensions to
arbitrary convex sets.

######  Theorem 5.9.

Let $K\subset V$ be a nonempty convex set. Then the family

$U=\left\\{\operatorname{ri}(F)\colon\text{$F$ is a face of
$K$}\right\\}\setminus\\{\emptyset\\}=\left\\{\operatorname{ri}(F_{K}(x))\colon
x\in K\right\\}$

is a partition of $K$. Every relative open convex subset $C$ of $K$ is
contained in one of the sets in $U$, namely $C\subset F_{K}(x)$ holds for
every $x\in C$. The sets in $U$ are the maximal relative open convex subsets
of $K$.

###### Proof.

Coro. 4.4 shows that $U$ is a partition. Let $C$ be a relative open convex
subsets of $K$ and let $x\in C$. Then Coro. 5.8 shows that
$\operatorname{ri}(F_{K}(x))$ includes $C$. Thm. 5.5 shows that
$\operatorname{ri}(F_{K}(x))$ is a relative open convex set. If
$\operatorname{ri}(F_{K}(x))$ is included in another relative open convex
subset $D\subset K$ then $D\subset F_{K}(x)$ follows, again from Coro. 5.8, as
$x\in\operatorname{ri}(F_{K}(x))$ holds by Coro. 3.4. ∎

######  Corollary 5.10.

A nonempty convex set $K$ is relative open if and only if it has exactly two
faces, which are $K$ and $\emptyset$.

###### Proof.

If $K$ is relative open and nonempty, then the partition in Thm. 5.9 is
$U=\\{K\\}$. Rem. 4.5.2 then shows that $K$ has at most two faces, $K$ and
$\emptyset$. Conversely, if $K$ has only one nonempty face, then
$\\{\operatorname{ri}(K)\\}$ is a partition of $K$ by Coro. 4.4, which implies
$K=\operatorname{ri}(K)$. ∎

##  6\. Novel results: Convex generators of faces

This section is motivated by, but not equivalent to statements from [7]. The
difference is due to nonequivalent concepts of a “face” (see Sec. 7).

Thm. 6.1 matches [7, (3.1)].

######  Theorem 6.1.

If $C\subset K$ is convex, then $\bigcup_{x\in C}F_{K}(x)$ is a face of $K$.

###### Proof.

The set $E=\bigcup_{x\in C}F_{K}(x)$ is a union of extreme sets, and hence an
extreme set itself. We complete the proof by showing that $E$ is convex.

Let $a_{1},a_{2}\in E$. There $c_{i}\in C$ such that $a_{i}\in F_{K}(c_{i})$
and Coro. 3.2 shows that there is $b_{i}\in K$ and $\eta_{i}\in(0,1)$ such
that $c_{i}=(1-\eta_{i})a_{i}+\eta_{i}b_{i}$, for $i=1,2$. Consider the set
$D$ defined as the convex hull of the points $a_{1},a_{2},b_{1},b_{2}$. We
lift these four points to some affinely independent points
$a_{1}^{\prime},a_{2}^{\prime},b_{1}^{\prime},b_{2}^{\prime}$ in
$\mathbb{R}^{3}$, which are the vertices of a tetrahedron. Let
$c=\frac{1}{2}(c_{1}+c_{2})$ and

$\textstyle
c^{\prime}=\frac{1}{2}\left((1-\eta)(a_{1}^{\prime}+a_{2}^{\prime})+\eta(b_{1}^{\prime}+b_{2}^{\prime})\right)\,\text{.}$

Clearly, the point $c^{\prime}$ lies in the relative interior of the
tetrahedron. And the point $c$ lies in the relative interior of $D$, because a
linear image of the relative interior of the tetrahedron is the relative
interior of the linear image of the whole tetrahedron [10, Thm. 6.6 ]. The
inclusions

$\textstyle[a_{1},a_{2}]\subset D\subset F_{K}(c)\subset E$

complete the proof that $E$ is convex. Thereby, the second inclusion follows
from Lemma 4.1.1, as $c$ is a relative interior point of $D$. The third
inclusion holds because $c\in C$. ∎

We define the _face generated by a subset $S\subset K$_ as the smallest face
of $K$ that contains $S$. We denote this face by $F_{K}(S)$.

Coro. 6.2 matches [7, (3.3)].

######  Corollary 6.2.

Let $S\subset K$. Then the face of $K$ generated by $S$ is the union
$F_{K}(S)=\bigcup_{x\in C}F_{K}(x)$, where $C$ is the convex hull of $S$.

###### Proof.

The union $F=\bigcup_{x\in C}F_{K}(x)$ is a face of $K$ by Thm. 6.1. Let $G$
be any face containing $S$. As $G$ is convex it contains $C$. The face $G$
also contains the face $F_{K}(x)$ for all $x\in C$ by Lemma 4.1.1, because
$x\in\operatorname{ri}(F_{K}(x))$ holds by Coro. 3.4. This proves $F\subset
G$. ∎

Coro. 6.3 matches [7, (4.7)].

######  Corollary 6.3.

Let $K,L\subset V$ be convex sets and let $K$ be relative open. Then every
extreme set resp. face of $K\cap L$ is the intersection of $K$ and an extreme
set resp. face of $L$.

###### Proof.

Let $E$ be an extreme set of $K\cap L$. Then Coro. 4.3.3 and Prop. 4.7.1 show

$\textstyle E=\bigcup_{x\in E}F_{K\cap L}(x)=\textstyle\bigcup_{x\in
E}\left(F_{K}(x)\cap F_{L}(x)\right)\,\text{.}$

As $K$ is relative open, Lemma 4.1.3 implies $F_{K}(x)=K$ for all $x\in E$,
hence

$\textstyle E=\textstyle\bigcup_{x\in E}\left(K\cap F_{L}(x)\right)=\textstyle
K\cap\bigcup_{x\in E}F_{L}(x)\,\text{.}$

The set $\bigcup_{x\in E}F_{L}(x)$ is a union of extreme sets of $L$ and hence
an extreme set of $L$ itself. If $E$ is a face of $K\cap L$, then $E$ is
convex and Thm. 6.1 completes the proof. ∎

##  7\. L. E. Dubins’s terminology

A _d-extreme set_ of $K$ is a subset of $K$ that contains all open segments
with endpoints in $K$ that it intersects. A _d-face_ of $K$ is a convex
d-extreme set of $K$. A point $x\in K$ is a _d-extreme point_ of $K$ if
$\\{x\\}$ is a d-face of $K$. Clearly, any union or intersection of d-extreme
sets of $K$ is a d-extreme set of $K$. Since any intersection of convex sets
is a convex set, any intersection of d-faces of $K$ is a d-face of $K$. Hence,
the intersection of all d-faces containing a given point $x\in K$ is a d-face
of $K$. This is the smallest d-face of $K$ that contains $x$.

The term “face” is used in [7] for what we call a d-face. It is interesting to
compare Thm. 7.1 with Coro. 4.3 and Rem. 4.5 above.

######  Theorem 7.1.

A subset $C\subset K$ is a d-extreme set of $K$ if and only if $C$ is a union
(possibly empty) of sets from the family

$U=\left\\{\operatorname{ri}(F_{K}(x))\colon x\in
K\right\\}=\left\\{\operatorname{ri}(F)\colon\text{$F$ is a face of
$K$}\right\\}\setminus\\{\emptyset\\}\,\text{.}$

If $E$ is a d-extreme set of $K$, then $E=\bigcup_{x\in
E}\operatorname{ri}(F_{K}(x))$. A subset $C\subset K$ is a d-face of $K$ if
and only if $C$ is a convex union of sets from the family $U$. In particular,
all the sets in $U$ are d-faces. A point $x\in K$ is a d-extreme point of $K$
if and only if $x$ is an extreme point of $K$.

###### Proof.

The two families describing $U$ are indeed equal by Coro. 4.4. Let $E$ be a
d-extreme set of $K$ and let $x\in E$. As
$x\in\operatorname{ri}(F_{K}(x))\subset E$ holds by Lemma 5.2, we have
$E=\bigcup_{x\in E}\operatorname{ri}(F_{K}(x))$. Conversely, assume an open
segment $(a,b)$ with endpoints $a,b\in K$ intersects a union of sets from the
family $U$. Say $x\in(a,b)\cap\operatorname{ri}(F)$ for a face $F$ of $K$.
Then Lemma 4.1.3 shows $F=F_{K}(x)$ and Coro. 5.8 implies that
$\operatorname{ri}(F_{K}(x))$, and hence the union, contains $(a,b)$, because
$(a,b)$ is a relative open convex set.

The characterization of d-faces follows from that of d-extreme sets. The sets
in $U$ are d-faces as they are convex by Lemma 4.2.

That “d-extreme point” and “extreme point” are equivalent terms is implied by
the fact that improper segments are singletons no matter whether they are open
segments or closed segments. ∎

We provide an alternative proof of [7, Thm. 2.1] in Coro. 7.2 below. As
observed in the introduction, a point $x\in K$ is an _internal point_ of $K$
if and only if $x$ is a relative interior point of $K$. By definition [7], a
convex set $K$ is an _internal convex set_ if every point of $K$ is an
internal point of $K$. Clearly, this is equivalent to $K$ being a relative
open convex set.

######  Corollary 7.2 (Dubins).

The smallest d-face of $K$ that contains a point $x\in K$ is equal to
$\operatorname{ri}(F_{K}(x))$ and to the largest internal convex subset of $K$
that contains $x$.

###### Proof.

As $x\in\operatorname{ri}(F_{K}(x))$ holds by Coro. 3.4, Thm. 7.1 shows that
$\operatorname{ri}(F_{K}(x))$ is the smallest d-face of $K$ that contains $x$.
As discussed in the paragraph above, the largest internal convex subset of $K$
that contains $x$ is the largest relative open convex subset of $K$ that
contains $x$. This set equals $\operatorname{ri}(F_{K}(x))$ by Thm. 5.9, since
the singleton $\\{x\\}$ is relative open and convex. ∎

An _elementary face_ [7] of $K$ is a d-face of $K$ that is an internal convex
set. In our terminology, an elementary face is a relative open d-face.

######  Corollary 7.3.

A nonempty subset of $K$ is an elementary face of $K$ if and only if it equals
$\operatorname{ri}(F_{K}(x))$ for some $x\in K$.

###### Proof.

The set $\operatorname{ri}(F_{K}(x))$ is a d-face by Thm. 7.1 and it is
relative open by Thm. 5.5 for all $x\in K$. Conversely, let $F$ be an
elementary face of $K$. Then

$\textstyle\bigcup_{x\in
F}\operatorname{ri}(F_{K}(x))=F\subset\operatorname{ri}(F_{K}(y))\quad\text{for
all $y\in F$.}$

The equality on the left holds by Thm. 7.1 as $F$ is a d-face. The inclusion
on the right holds by Thm. 5.9 as $F$ is a relative open convex set. This
chain of inclusions implies that $F=\operatorname{ri}(F_{K}(x))$ holds for all
$x\in F$. Unless $F=\emptyset$, this means that we have
$F=\operatorname{ri}(F_{K}(x))$ for some $x\in F$. ∎

##  8\. Example 1: Spaces of probability measures

Let $\mathcal{P}=\mathcal{P}(\Omega,\mathcal{A})$ denote the convex set of
probability measures on a measurable space $(\Omega,\mathcal{A})$. A
probability measure $\lambda\in\mathcal{P}$ is _absolutely continuous_ with
respect to $\mu\in\mathcal{P}$, symbolically $\lambda\ll\mu$, if every
$\mu$-null set is a $\lambda$-null set. The measures are _equivalent_ ,
$\lambda\equiv\mu$, if $\lambda\ll\mu$ and $\mu\ll\lambda$. If
$\lambda\ll\mu$, then we denote by
$\frac{\operatorname{\mathrm{d}}\lambda}{\operatorname{\mathrm{d}}\mu}:\Omega\to[0,\infty)$,
the _Radon-Nikodym derivative_ of $\lambda$ with respect to $\mu$, wich is a
measurable function satisfying
$\lambda(A)=\int_{A}\frac{\operatorname{\mathrm{d}}\lambda}{\operatorname{\mathrm{d}}\mu}\,\operatorname{\mathrm{d}}\mu$
for all $A\in\mathcal{A}$, see for example Halmos [8, Sec. 31].

If $\mu\in\mathcal{P}$, then we say a proposition $\pi(\omega)$,
$\omega\in\Omega$, is true _$\mu$ -almost surely_, which we abbreviate as
_$\mu$ -a.s._, if $\mu(\\{\omega\in\Omega\colon\text{$\pi(\omega)$ is
false}\\})=0$.

######  Theorem 8.1.

Let $\lambda,\mu\in\mathcal{P}$. The following assertions are equivalent.

  1. 1)

The measure $\lambda$ lies in the face $F_{\mathcal{P}}(\mu)$ of $\mathcal{P}$
generated by $\mu$.

  2. 2)

There is $c\in(0,\infty)$ such that $\lambda(A)\leq c\,\mu(A)$ holds for all
$A\in\mathcal{A}$.

  3. 3)

We have $\lambda\ll\mu$ and there is $c\in(0,\infty)$ such that
$\frac{\operatorname{\mathrm{d}}\lambda}{\operatorname{\mathrm{d}}\mu}\leq c$
holds $\mu$-a.s..

###### Proof.

Thm. 3.1 shows that a probability measure $\lambda\in\mathcal{P}$ lies in
$F_{\mathcal{P}}(\mu)$ if and only if there is $\epsilon>0$ such that
$\mu+\epsilon(\mu-\lambda)\in\mathcal{P}$. The latter condition is equivalent
to the nonnegativity of the set function $\mu+\epsilon(\mu-\lambda)$, and
hence to part 2) of the theorem. It remains to prove the equivalence
2)$\Leftrightarrow$3).

If $\lambda\ll\mu$ and if there is $c>1$ such that
$\frac{\operatorname{\mathrm{d}}\lambda}{\operatorname{\mathrm{d}}\mu}(\omega)\leq
c$ holds $\mu$-a.s., then part 2) follows (with the same constant $c$),

$\textstyle\lambda(A)=\int_{A}\frac{\operatorname{\mathrm{d}}\lambda}{\operatorname{\mathrm{d}}\mu}\,\operatorname{\mathrm{d}}\mu\leq
c\,\mu(A)\quad\text{for all $A\in\mathcal{A}$.}$

Conversely, if $\lambda\ll\mu$ is false, then there is $A\in\mathcal{A}$ such
that $\lambda(A)>\mu(A)=0$, making part 2) impossible. If $\lambda\ll\mu$ is
true but
$\frac{\operatorname{\mathrm{d}}\lambda}{\operatorname{\mathrm{d}}\mu}$ is not
bounded $\mu$-a.s., then for every $c>0$ there is $A\in\mathcal{A}$ such that
$\mu(A)>0$ and
$\frac{\operatorname{\mathrm{d}}\lambda}{\operatorname{\mathrm{d}}\mu}(\omega)>c$
holds for all $\omega\in A$. Then

$\textstyle\lambda(A)=\int_{A}\frac{\operatorname{\mathrm{d}}\lambda}{\operatorname{\mathrm{d}}\mu}\,\operatorname{\mathrm{d}}\mu>c\,\mu(A)$

proves that part 2) fails. ∎

Thm. 8.1 and Coro. 8.2 generalize [13, Lemma 2.12] from a countable state
space to an arbitrary measurable space.

######  Corollary 8.2.

Let $\lambda,\mu\in\mathcal{P}$. The following assertions are equivalent.

  1. 1)

The measure $\lambda$ lies in $\operatorname{ri}(F_{\mathcal{P}}(\mu))$.

  2. 2)

There is $c>1$ such that $\mu(A)/c\leq\lambda(A)\leq c\,\mu(A)$ for all
$A\in\mathcal{A}$.

  3. 3)

We have $\lambda\equiv\mu$ and there are $c_{1},c_{2}\in(1,\infty)$ such that
$\frac{\operatorname{\mathrm{d}}\lambda}{\operatorname{\mathrm{d}}\mu}\leq
c_{1}$ holds $\mu$-a.s. and
$\frac{\operatorname{\mathrm{d}}\mu}{\operatorname{\mathrm{d}}\lambda}\leq
c_{2}$ holds $\lambda$-a.s..

  4. 4)

We have $\lambda\ll\mu$ and there is $c\in(1,\infty)$ such that
$\frac{1}{c}\leq\frac{\operatorname{\mathrm{d}}\lambda}{\operatorname{\mathrm{d}}\mu}\leq
c$ holds $\mu$-a.s..

###### Proof.

Lemma 5.1 shows that $\lambda\in\operatorname{ri}(F_{\mathcal{P}}(\mu))$ is
equivalent to $\lambda\in F_{\mathcal{P}}(\mu)$ and $\mu\in
F_{\mathcal{P}}(\lambda)$, so the equivalences
1)$\Leftrightarrow$2)$\Leftrightarrow$3) follow from those of Thm. 8.1.

Note that $\lambda$-a.s. is the same as $\mu$-a.s. if $\lambda\equiv\mu$.
Hence, 3) implies that
$\frac{\operatorname{\mathrm{d}}\mu}{\operatorname{\mathrm{d}}\lambda}\cdot\frac{\operatorname{\mathrm{d}}\lambda}{\operatorname{\mathrm{d}}\mu}=1$
and hence
$\frac{\operatorname{\mathrm{d}}\lambda}{\operatorname{\mathrm{d}}\mu}=(\frac{\operatorname{\mathrm{d}}\mu}{\operatorname{\mathrm{d}}\lambda})^{-1}\geq
1/c_{2}$ holds $\mu$-a.s., see for example [8, Thm. A, p. 133]. Conversely, if
$\lambda\ll\mu$ and if there is $c\in(1,\infty)$ such that
$\frac{1}{c}\leq\frac{\operatorname{\mathrm{d}}\lambda}{\operatorname{\mathrm{d}}\mu}$
holds $\mu$-a.s., then

$\textstyle\lambda(A)=\int_{A}\frac{\operatorname{\mathrm{d}}\lambda}{\operatorname{\mathrm{d}}\mu}\,\operatorname{\mathrm{d}}\mu\geq\frac{1}{c}\,\mu(A)\quad\text{for
all $A\in\mathcal{A}$}$

implies that $\mu\equiv\lambda$ and that
$\frac{\operatorname{\mathrm{d}}\mu}{\operatorname{\mathrm{d}}\lambda}=(\frac{\operatorname{\mathrm{d}}\lambda}{\operatorname{\mathrm{d}}\mu})^{-1}\leq
c$ holds $\lambda$-a.s.. ∎

An example in [7] mentions that the smallest d-face of $\mathcal{P}$ that
contains a probability measure $\mu\in\mathcal{P}$, consists of those
$\lambda\in\mathcal{P}$ for which there is $c>0$, such that for all
$A\in\mathcal{A}$ we have $\mu(A)\leq c\,\lambda(A)\leq c^{2}\mu(A)$. Coro.
7.2 translates this assertion into the equivalence 1)$\Leftrightarrow$2) of
Coro. 8.2.

##  9\. Example 2: Convex cores

In a second example, we consider the set of probability measures on the Borel
$\sigma$-algebra $\mathcal{B}(d)$ of $\mathbb{R}^{d}$. The _convex core_
$\operatorname{cc}(\mu)$ of
$\mu\in\mathcal{P}=\mathcal{P}(\mathbb{R}^{d},\mathcal{B}(d))$ is the
intersection of all convex sets $C\in\mathcal{B}(d)$ of full measure
$\mu(C)=\mu(\mathbb{R}^{d})$. The convex core was introduced in [6] to extend
exponential families in a natural way, such that information projections
become properly defined. The _mean_ of $\mu$ is the integral
$m(\mu)=\int_{\mathbb{R}^{d}}x\operatorname{\mathrm{d}}\mu(x)\in\mathbb{R}^{d}$,
provided that each coordinate function is $\mu$-integrable; otherwise, $\mu$
does not have a mean.

######  Theorem 9.1 (Csiszár and Matúš).

The convex core of $\mu\in\mathcal{P}(\mathbb{R}^{d},\mathcal{B}(d))$ equals
$\operatorname{cc}(\mu)=m\left(\left\\{\lambda\in\mathcal{P}\colon\lambda\ll\mu\right\\}\right)$.
Moreover, to each $a\in\operatorname{cc}(\mu)$ there exists
$\lambda\in\mathcal{P}$ with $\lambda\ll\mu$ and mean $m(\lambda)=a$ such that
$\frac{\operatorname{\mathrm{d}}\lambda}{\operatorname{\mathrm{d}}\mu}$ is
bounded $\mu$-a.s..

Thm. 9.1 is proved in Thm. 3 of [6]. We derive from it a description of the
relative interior of the convex core.

######  Corollary 9.2.

The convex core of $\mu\in\mathcal{P}(\mathbb{R}^{d},\mathcal{B}(d))$ is
$\operatorname{cc}(\mu)=m(F_{\mathcal{P}}(\mu))$. The relative interior is
$\operatorname{ri}(\operatorname{cc}(\mu))=m(\operatorname{ri}(F_{\mathcal{P}}(\mu)))$,
which equals

$\textstyle\operatorname{ri}(\operatorname{cc}(\mu))=m\left(\left\\{\lambda\in\mathcal{P}\mid\lambda\equiv\mu,\exists
c\in(1,\infty)\colon\frac{1}{c}\leq\frac{\operatorname{\mathrm{d}}\lambda}{\operatorname{\mathrm{d}}\mu}\leq
c\mbox{\leavevmode\nobreak\ $\mu$-a.s.}\right\\}\right)\,\text{.}$

###### Proof.

Thm. 8.1 and Thm. 9.1 show that
$\operatorname{cc}(\mu)=m(F_{\mathcal{P}}(\mu))$. As $\mu$ is a relative
interior point of $F_{\mathcal{P}}(\mu)$ by Coro. 3.4, we obtain

$\operatorname{ri}(\operatorname{cc}(\mu))=m(\operatorname{ri}(F_{\mathcal{P}}(\mu)))$

from Thm. 5.7. Coro. 8.2 completes the proof. ∎

The characterization of $\operatorname{ri}(\operatorname{cc}(\mu))$ in Coro.
9.2 is somewhat stronger than that in Lemma 5 of [6], which ignores the lower
bound
$0<\frac{1}{c}\leq\frac{\operatorname{\mathrm{d}}\lambda}{\operatorname{\mathrm{d}}\mu}$
$\mu$-a.s.. Lemma 5 of [6] also shows
$\operatorname{ri}(\operatorname{cc}(\mu))=m\left(\left\\{\lambda\in\mathcal{P}\colon\lambda\equiv\mu\right\\}\right)$,
which cannot be deduced from Thm. 9.1 with the methods developed above,
without the assistance of other methods.

##  10\. Example 3: Discrete probability measures

In a third example, we consider the set of probability measures on the
discrete $\sigma$-algebra $2^{\mathbb{N}}$ of $\mathbb{N}$.

A probability measure
$\mu\in\mathcal{P}=\mathcal{P}(\mathbb{N},2^{\mathbb{N}})$ is equivalently
described by its probability mass function
$p=\frac{\operatorname{\mathrm{d}}\mu}{\operatorname{\mathrm{d}}\nu}$ with
respect to the counting measure $\nu$, where $\nu(A)=\infty$ if the
cardinality of $A\subset\mathbb{N}$ is infinite, and $\nu(A)$ is the
cardinality of $A$ otherwise. The set of probability measures $\mathcal{P}$ is
affinely isomorphic to

$\textstyle\Delta_{\mathbb{N}}=\left\\{p:\mathbb{N}\to\mathbb{R}\mid\forall
n\in\mathbb{N}\colon p(n)\geq 0\mbox{\leavevmode\nobreak\
and\leavevmode\nobreak\ }\sum_{n=1}^{\infty}p(n)=1\right\\}\,\text{.}$

The _support_ of $p\in\Delta_{\mathbb{N}}$ is
$\operatorname{supp}(p)=\\{n\in\mathbb{N}\colon p(n)>0\\}$. For
$I\subset\mathbb{N}$ we consider the following sets of probability mass
functions

$\displaystyle\Delta_{I}$
$\displaystyle=\\{p\in\Delta_{\mathbb{N}}\colon\operatorname{supp}(p)\subset
I\\}\,\text{,}$ $\displaystyle\Delta_{I,\text{fin}}$
$\displaystyle=\\{p\in\Delta_{I}\colon\nu(\operatorname{supp}(p))<\infty\\}\,\text{.}$

The sets $\Delta_{I}$ and $\Delta_{I,\text{fin}}$, $I\subset\mathbb{N}$, are
faces of $\Delta_{\mathbb{N}}$. They have no relative interior points if
$\nu(I)=\infty$. The question was raised [13, Sec. 2] as to whether there
exist other faces of $\Delta_{\mathbb{N}}$ without relative interiors, too. It
turns out yes. Before giving examples, we characterize faces generated by
points.

######  Lemma 10.1.

For $p\in\Delta_{\mathbb{N}}$ we have

$\displaystyle\textstyle F_{\Delta_{\mathbb{N}}}(p)$
$\displaystyle=\textstyle\left\\{q\in\Delta_{\operatorname{supp}(p)}\colon\sup_{n\in\operatorname{supp}(p)}q(n)/p(n)<\infty\right\\}\,\text{,}$
$\displaystyle\operatorname{ri}(F_{\Delta_{\mathbb{N}}}(p))$
$\displaystyle=\textstyle\left\\{q\in
F_{\Delta_{\mathbb{N}}}(p)\colon\inf_{n\in\operatorname{supp}(p)}q(n)/p(n)>0\right\\}\,\text{.}$

###### Proof.

A proof is given in [13, Lemma 2.12]. The claim follows also from Thm. 8.1 and
Coro. 8.2. ∎

Example 10.2 and Lemma 10.3 show that the sets $\Delta_{I}$ and
$\Delta_{I,\text{fin}}$, $I\subset\mathbb{N}$, are not the only faces of
$\Delta_{\mathbb{N}}$ without relative interior points.

######  Example 10.2.

We consider the probability mass function $p_{s}:\mathbb{N}\to\mathbb{R}$,
$n\mapsto\zeta(s)^{-1}\cdot n^{-s}$ for all $s>1$, where
$\zeta(s)=\sum_{n\in\mathbb{N}}n^{-s}$ is the Euler-Riemann zeta function.
Lemma 10.1 shows that $p_{s}\in F_{\Delta_{\mathbb{N}}}(p_{t})$ if and only if
$t\leq s$. Hence, Lemma 4.1.2 proves

$F_{\Delta_{\mathbb{N}}}(p_{s})\subseteq F_{\Delta_{\mathbb{N}}}(p_{t})\iff
t\leq s,\qquad s,t>1.$

According to Lemma 10.3 below, any of the unions

$\textstyle F=\bigcup_{t>s}F_{\Delta_{\mathbb{N}}}(p_{t})\,\text{,}\quad s\geq
1\,\text{,}$

is a face of $\Delta_{\mathbb{N}}$ and $\operatorname{ri}(F)=\emptyset$. If
$s>1$, then the proper inclusion $F\subset F_{\Delta_{\mathbb{N}}}(p_{s})$
shows $F\neq\Delta_{\mathbb{N}}$. Hence $F$ is not of the form $\Delta_{I}$ or
$\Delta_{I,\text{fin}}$ for any $I\subset\mathbb{N}$.

######  Lemma 10.3.

Let $\\{x_{\alpha}\\}_{\alpha\in A}$ be a set of points in a convex set $K$
indexed by a totally ordered set $A$ without a maximal element in $A$, such
that $\alpha\leq\beta$ holds if and only if $F_{K}(x_{\alpha})\subset
F_{K}(x_{\beta})$ holds for all $\alpha,\beta\in A$. Then

$\textstyle F=\bigcup_{\alpha\in A}F_{K}(x_{\alpha})$

is a face of $K$ and $\operatorname{ri}(F)=\emptyset$.

###### Proof.

The set $F$ is convex. If $a,b\in F$, then $a\in F_{K}(x_{\alpha})$ and $b\in
F_{K}(x_{\alpha})$ for some $\alpha,\beta\in A$. Both points $x_{\alpha}$ and
$x_{\beta}$ lie in $F_{K}(x_{\max(\alpha,\beta)})$, hence the closed segment
with endpoints $a,b$ lies in $F_{K}(x_{\max(\alpha,\beta)})\subset F$. The set
$F$ is an extreme set. If the open segment with endpoints $a,b\in K$
intersects $F$, then it intersects $F_{K}(x_{\alpha})$ for some $\alpha\in A$.
It follows that $a,b\in F_{K}(x_{\alpha})\subset F$.

If $a\in\operatorname{ri}(F)$, then there would be $\alpha\in A$ with $a\in
F_{K}(x_{\alpha})$. Lemma 4.1.1 would then imply $F\subset F_{K}(x_{\alpha})$,
which is a contradiction. ∎

Whereas the set $\Delta_{\mathbb{N}}$ of all probability mass functions is a
closed subset of the Banach space [14]

$\textstyle\ell^{1}=\\{x:\mathbb{N}\to\mathbb{C}\mid\sum_{n=1}^{\infty}|x(n)|<\infty\\}$

with respect to the $\ell^{1}$-norm $\|x\|_{1}=\sum_{n=1}^{\infty}|x(n)|$,
many of its faces are not closed. Note that the $\ell_{1}$-norm on
$\Delta_{\mathbb{N}}$ corresponds to the _distance in variation_ between
$\lambda,\mu\in\mathcal{P}$,

$\textstyle\|\mu-\lambda\|:=2\sup_{A\subset\mathbb{N}}|\mu(A)-\lambda(A)|=\|p-q\|_{1}\,\text{,}$

where $p=\frac{\operatorname{\mathrm{d}}\mu}{\operatorname{\mathrm{d}}\nu}$
and $q=\frac{\operatorname{\mathrm{d}}\lambda}{\operatorname{\mathrm{d}}\nu}$
are the probability mass functions [12, Sec. 3.9]. Let $e_{n}\in\ell^{1}$ be
defined by $e_{n}(m)=1$ if $n=m$ and by $e_{n}(m)=0$ if $n\neq m$.

######  Lemma 10.4.

A face $F$ of $\Delta_{\mathbb{N}}$ is closed in $\Delta_{\mathbb{N}}$ in the
$\ell^{1}$-norm if and only if $F=\Delta_{I}$ for some $I\subset\mathbb{N}$.

###### Proof.

To begin with, the face $\Delta_{I,\text{fin}}$ of functions with finite
support is dense in $\Delta_{I}$ for all $I\subset\mathbb{N}$. To see this,
let $I=\mathbb{N}$ (without loss of generality) and let
$p\in\Delta_{\mathbb{N}}$. Then
$(p_{k})_{k\in\mathbb{N}}\subset\Delta_{\mathbb{N},\text{fin}}$, defined by

$p_{k}(n)=\left\\{\begin{array}[]{ll}p(n)&\text{if $n<k$,}\\\ \sum_{m\geq
k}p(m)&\text{if $n=k$,}\\\ 0&\text{else,}\end{array}\right.\quad
k,n\in\mathbb{N}\,\text{,}$

converges to $p$, as $\|p-p_{k}\|_{1}=2\sum_{m>k}p(m)$ for all
$k\in\mathbb{N}$.

Second, if $F$ is a face of $\Delta_{\mathbb{N}}$, then
$\Delta_{I,\text{fin}}\subset F$ holds for $I=\bigcup_{p\in
F}\operatorname{supp}(p)$. To see this, let $n\in I$, and let $p\in F$ such
that $n\in\operatorname{supp}(p)$. As
$p\in\operatorname{ri}(F_{\Delta_{\mathbb{N}}}(p))$ holds by Coro. 3.4, we get
$e_{n}\in F_{\Delta_{\mathbb{N}}}(p)$ from Coro. 3.2. Then $e_{n}\in F$
follows from Lemma 4.1.1. We have proved that $e_{n}\in F$ for all $n\in I$.
This implies $\Delta_{I,\text{fin}}\subset F$ as claimed.

The preceding arguments show that the closure of any face $F$ of
$\Delta_{\mathbb{N}}$ is $\Delta_{I}$ where $I=\bigcup_{p\in
F}\operatorname{supp}(p)$. The converse, that the faces $\Delta_{I}$,
$I\subset\mathbb{N}$, are closed can be shown with methods from elementary
functional analysis [14]. ∎

Note that the result of Lemma 10.4, that the norm closed faces of
$\Delta_{\mathbb{N}}$ are in a one-to-one correspondence with the subsets of
$\mathbb{N}$ is just a special case of a more general property of von Neumann
algebras [2]. The space $\ell^{1}$ is the predual of the von Neumann algebra

$\textstyle\ell^{\infty}=\\{x:\mathbb{N}\to\mathbb{C}\mid\sup_{n\in\mathbb{N}}|x(n)|<\infty\\}\,\text{.}$

The set $\Delta_{\mathbb{N}}\subset\ell^{1}$ is the _normal state space_ of
$\ell^{\infty}$, and the subsets $A$ of $\mathbb{N}$ are in a one-to-one
correspondence with the projections $\sum_{n\in A}e_{n}$ in $\ell^{\infty}$.
In a general von Neumann algebra, there is an order preserving isomorphism
between the norm closed faces of the normal state space and the projections in
the algebra [2, Thm. 3.35]. Moreover, a characterization of the closure of the
face generated by a normal state is provided in [2, Coro. 3.28].

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Stephan Weis

Berlin, Germany

e-mail<EMAIL_ADDRESS>