# Proton Radius: A Puzzle or a Solution!?

Ulrich D. Jentschura  Department of Physics and LAMOR, Missouri University of
Science and Technology, Rolla, Missouri 65409, USA  MTA–DE Particle Physics
Research Group, P.O. Box 51, H-–4001 Debrecen, Hungary  MTA Atomki, P.O. Box
51, H–4001 Debrecen, Hungary

###### Abstract

The proton radius puzzle is known as the discrepancy of the proton radius,
obtained from muonic hydrogen spectroscopy (obtained as being roughly equal to
$0.84$ fm), and the proton radius obtained from (ordinary) hydrogen
spectroscopy where a number of measurements involving highly excited states
have traditionally favored a value of about $0.88$ fm. Recently, a number of
measurements of hydrogen transitions by the Munich (Garching) groups (notably,
several hyperfine-resolved sublevels of the $2S$–$4P$) and by the group at the
University of Toronto ($2S$–$2P_{1/2}$) have led to transition frequency data
consistent with the smaller proton radius of about $0.84$ fm. A recent
measurement of the $2S$–$8D$ transition by a group at Colorado State
University leads to a proton radius of about $0.86$ fm, in between the two
aforementioned results. The current situation points to a possible, purely
experimental, resolution of the proton radius puzzle. However, a closer look
at the situation reveals that the situation may be somewhat less clear,
raising the question of whether or not the proton radius puzzle has been
conclusively solved, and opening up interesting experimental possiblities at
TRIUMF/ARIEL.

##  I Introduction

We present a brief account of the current status of measurements of the proton
radius, based on recent hydrogen spectroscopy BeEtAl1997 ; ScEtAl1999 ;
JeKoLBMoTa2005 ; PoEtAl2010 ; Je2015muonic ; BeEtAl2017 ; BeEtAl2019 ;
BrEtAl2022  and scattering experiments BeEtAl2010 ; XiEtAl2019 . Also, we
point out that a current discrepancy of the deuteron radius, based on
deuterium spectroscopy and scattering experiments, merits further
investigation PoEtAl2016 . Attention is drawn to the fact that in 1969,
coindicidentally on the same day on which mankind set foot on the moon (21st
of July, 1969), two Letters were published CaEtAl1969prl1 ; CaEtAl1969prl2 ,
which report on an observation of a larger slope of the Sachs form factor of
the proton, when measured using electron versus muon scattering, consistent
with a (roughly) 2% difference in the proton radii derived from either
scattering method. The current situation opens interesting experimental
possibilities for the MUSE experiment MUSE ; KoEtAl2014 ; Ko2022priv  at the
Paul–Scherrer Institute (PSI) and for a potential muon option at the Advanced
Rare Isotope Laboratory (ARIEL) at TRIUMF (Vancouver).

##  II Brookhaven Experiment

It might be useful to recall that scattering experiments on proton targets,
using either electron or proton projectiles, have a considerable history. One
example dates from about five decades ago CaEtAl1969prl1 ; CaEtAl1969prl2 . It
was observed that muon-proton ($\mu$-$p$) versus electron-proton ($e$-$p$)
scattering data for the Sachs form factor taken in the range $0.15\,({\rm
GeV}/c)^{2}<q^{2}<0.85\,({\rm GeV}/c)^{2}$ differ from each other by a
relative factor $N=0.960\pm 0.006$. It is stated, in the right text column on
p. 154 of Ref. CaEtAl1969prl1 , that the “4% suppression of the form-factor
ratio represents an 8% difference in the cross section since
$\mathrm{d}\sigma/\mathrm{d}q^{2}\propto G^{2}$”.

One should note that the authors of Ref. CaEtAl1969prl2  base their analysis
on a number of assumptions. Let us consider the Rosenbluth formula [see Eq.
(1) of Ref. CaEtAl1969prl1 ],

$\left.\frac{\mathrm{d}\sigma}{\mathrm{d}q^{2}}\right|_{\mu,e}=\left.\frac{\mathrm{d}\sigma}{\mathrm{d}q^{2}}\right|_{\rm
NS}\frac{1}{\cot^{2}(\theta/2)}\,\\\ \times\left[2\tau
G_{M}(q^{2})+\frac{G_{E}^{2}(q^{2})+\tau\,G_{M}(q^{2})}{1+\tau}\,\cot^{2}(\theta/2)\right]\,,$
(1)

where $\tau=-q^{2}/(4m_{p}^{2})=\vec{q}^{2}/(4m_{p}^{2})$ (for purely space-
like momentum transfer), $\theta$ is the scattering angle, and
$\left.\mathrm{d}\sigma/\mathrm{d}q^{2}\right|_{\rm NS}$ is the differential
cross section for spinless particles. The proton radius is inferred in Ref.
CaEtAl1969prl2  as follows [see also Eq. (11) of Ref. Je2011radii ],

$r_{p}=\sqrt{\langle r^{2}\rangle_{p}}\,,\qquad\langle
r^{2}\rangle_{p}=r_{p}^{2}=6\hbar^{2}\left.\frac{\partial
G_{E}(q^{2})}{\partial q^{2}}\right|_{q^{2}=0}$ (2)

The experiment CaEtAl1969prl1 ; CaEtAl1969prl2  has been carried out in a
region of small scattering angle $\theta$, where the second (slope) term in
square brackets in Eq. (1) dominates the first (intercept), and the electric
and magnetic form factors cannot be determined separately. The authors of of
Refs. CaEtAl1969prl2  base their analysis on the assumption that
$G_{E}=G_{M}/\mu\equiv G$, where $\mu\approx 1$ is the ratio of the magnetic
and electric form factors of the proton. This assumption is rather well
justified and is in fact confirmed by more recent experiments JoEtAl2000prl .

In the abstract of Ref. CaEtAl1969prl2 , it is stated that the “apparent
disagreement [of the $\mu$-$p$ versus $e$-$p$ data] can possibly be accounted
for by a combination of systematic normalization errors”. This statement
highlights a possible problem in the experiment CaEtAl1969prl1 ;
CaEtAl1969prl2 : Namely, it is experimentally difficult to accurately
normalize the flux of incoming particles, which could in principle account for
the tentative muon-proton nonuniversality observed in Refs. CaEtAl1969prl1 ;
CaEtAl1969prl2 . Yet, as evident from the rather detailed analysis present in
the two Letters CaEtAl1969prl1 ; CaEtAl1969prl2 , it is evident that
considerable efforts were made by the authors of Refs. CaEtAl1969prl1 ;
CaEtAl1969prl2  to avoid such normalization errors. In fact, in Ref.
CaEtAl1969prl1 , the scattering data were checked for the possible presence of
inelastic and multiple-scattering contributions, and also, the validity of the
functional form (1) was verified against experimental data, for both muon as
well as anti-muon projectiles. Let us also record an observation: The
differential cross section is roughly proportional to the square of the Sachs
electric $G_{E}$ form factor (under the assumptions made above), and the slope
of the Sachs form factor is again proportional to the square of the proton
radius. So, the observed (roughly) 8% difference of the $\mu$-$p$ scattering
cross sections CaEtAl1969prl1 ; CaEtAl1969prl2  translates into a 4%
difference of the form factor, which translates into a 2% difference of the
proton radii derived from $\mu$-$p$ versus $e$-$p$ scattering. Tentatively,
one could argue that, if the trend seen in the 1969 experiment were to be
exclusively due to a normalization error, the relative systematic error would
have to be on the order of 8% for the cross section, which is quite large.

Coincidentally, this is exactly the difference (both sign and magnitude agree)
between the proton radii derived from the most recent $2S$–$8D$ measurement on
atomic hydrogen BrEtAl2022  and the result from muonic hydrogen spectroscopy
PoEtAl2010 . This situation gives a very pronounced motivation for the MuSE
experiment MUSE ; KoEtAl2014 ; Ko2022priv , and also for adding a possible
muon option to TRIUMF/ARIEL.

Figure 1:  [ Reproduced with permission. ] Cross section data from Refs.
CaEtAl1969prl1 ; CaEtAl1969prl2 .

##  III Brief Review of Hydrogen Experiments

In contrast to scattering experiments, the proton radius is derived from
spectroscopic experiments via the finite-size energy shift, which, in leading
order and applying the nonrelativistic approximation, reads as

$\Delta E=\frac{2}{3}\,\frac{(Z\alpha)^{4}\,\mu
c^{2}}{\pi\,n^{3}}\,\delta_{\ell
0}\,\left(\frac{\mu\,c\,r_{p}}{\hbar}\right)^{2}\,,$ (3)

where $Z$ is the nuclear charge number of the hydrogenlike ion, $\alpha$ is
the fine-structure constant, $\mu$ is the reduced mass, $c$ is the speed of
light, and $\ell$ is the orbital angular momentum. The Kronecker $\delta$
implies that the energy correction is nonvanishing only for $S$ states with
$\ell=0$. The relation (3) is only the leading approximation to the finite-
size effect. Higher-order terms involve convoluted moments of the nuclear-
charge distribution, as discussed in Chap. 5 of Ref. JeAd2022book  and in
great detail in Ref. Fr1979 .

Let us discuss Eq. (3). We have $Z=1$ for both hydrogen and muonic hydrogen.
The reduced mass $\mu$ is roughly 200 times larger for muonic bound systems as
compared to ordinary hydrogen. The finite-size effect is proportional to
$\mu^{3}$, and thus, muonic hydrogen energy levels constitute a very sensitive
probe of the proton radius.

In chronological order, let us discuss some recent results for the proton
radius from spectroscopic BeEtAl1997 ; ScEtAl1999 ; JeKoLBMoTa2005 ;
PoEtAl2010 ; Je2015muonic ; BeEtAl2017 ; BeEtAl2019 ; BrEtAl2022 , as well as
scattering, experiments BeEtAl2010 ; XiEtAl2019 . We first recall that a
least-squares fit of all experimental data available up to 2005 for both
hydrogen as well as deuterium led to the results $r_{p}=0.8750(68)\,{\rm
fm}\approx 0.88\,{\rm fm}$ and $r_{d}=2.1394(28)\,{\rm fm}$, based on a total
of twenty-three measured transitions, including transitions to highly excited
states JeKoLBMoTa2005 .

In 2010, the muonic hydrogen $2S$–$2P_{1/2}$ Lamb shift measurement at PSI
(Ref. PoEtAl2010 ) led to a proton radius of $r_{p}=0.84184(67)\,{\rm
fm}\approx\,0.84\,{\rm fm}$. The result was in disagreement with the results
of an electron scattering experiment carried out in Mainz (in 2010, see Ref.
BeEtAl2010 ), which led to the result of $r_{p}=0.879(8)\,{\rm
fm}\approx\,0.88\,{\rm fm}$, consistent with the value obtained from atomic
hydrogen spectroscopys JeKoLBMoTa2005 .

Since 2010, the discrepancy between the proton radius obtained from hydrogen
spectroscopy ($r_{p}\approx 0.88\,{\rm fm}$) and from muonic hydrogen
spectroscopy ($r_{p}\approx 0.84\,{\rm fm}$) has been termed the proton radius
puzzle. In 2017, a measurement of the $2S$–$4P$ transition in atomic hydrogen
(Ref. BeEtAl2017 , taking into account various fine-structure and hyperfine
sublevels) led to a result of $r_{p}=0.8335(95)\,{\rm fm}$, consistent with
the smaller proton radius, otherwise obtained from muonic hydrogen
spectroscopy. In the analysis of the experiment BeEtAl2017 , emphasis is laid
on so-called cross-damping terms, which had been analyzed (for differential
cross sections) in Ref. JeMo2002 .

A recent hydrogen measurement (in 2018, by the French group) of the $1S$–$3S$
transition FlEtAl2018  is consistent with the larger proton radius, with a
result of $r_{p}=0.877(13)\,{\rm fm}\approx 0.88\,{\rm fm}$. In Ref.
Je2015muonic , it had been pointed out that the experimental approach taken by
the Paris group should be largely independent of cross-damping terms.

The PRad Scattering Experiment, in 2019, obtained a value of
$r_{p}=0.831(14)\,{\rm fm}$, as reported in Ref. XiEtAl2019 , based on
scattering data. In 2019, a spectroscopic measurement in Toronto, of the
$2S$–$2P_{1/2}$ Lamb shift in atomic hdrogen, resulted in a proton radius of
$r_{p}=0.833(10)\,{\rm fm}$ BeEtAl2019 , consistent with the smaller value of
the proton radius. One might point that the $2P_{1/2}$–$2P_{3/2}$ fine-
structure is nearly independent of the proton radius and can be calculated to
very high precision JePa1996 ; its measurement would constitute an important
consistency check for the smallness of the “Canadian protons” measured in Ref.
XiEtAl2019 . (Of course, the Einstein equivalence principle (EEP) implies that
the outcome of an experiment should be independent of where and when in the
Universe it is performed. Assuming that protons are indistinguishable,
identical elementary particles, one might be tempted to include the tongue-in-
the-cheek remark that any conceivable size difference in “French versus
Canadian and German Protons” would violate the EEP.)

In Ref. Je2015muonic , it had been pointed out that the so-called cross-
damping terms, which had played an important role in the analysis of the
$2S$–$4P$ experiment BeEtAl2017 , should be suppressed in $2S$–$nD$
transitions in atomic hydrogen [see the discussion surrounding Eqs. (28)—(31)
of Ref. Je2015muonic ]. It had also been pointed out in Ref. Je2015muonic
that cross-damping terms are “not likely to shift the experimental results
reported in Refs. BeEtAl1997  and ScEtAl1999  on a level commensurate with the
proton radius puzzle energy shift.” We recall that Refs. BeEtAl1997  and
ScEtAl1999  report on measurements of $2S$–$nD$ transitions with $n=8$ and
$n=12$. In consequence, it is of significance that a very recent (2022)
measurement of the $2S_{1/2}$–$8D_{5/2}$ transition in atomic hydrogen (see
Ref. BrEtAl2022 ) has led to a proton radius of $r_{p}=0.8584(51)\,{\rm
fm}\approx 0.86\,{\rm fm}$, which lies in between the previously accepted
value of $r_{p}\approx 0.88\,{\rm fm}$ from atomic hydrogen spectroscopy
JeKoLBMoTa2005  and the value of $r_{p}\approx 0.84\,{\rm fm}$ derived from
muonic hydrogen spectroscopy PoEtAl2010 .

The proton radius puzzle is exacerbated by the fact that the raw data for the
Sachs form factor derived from the 2010 Mainz experiment BeEtAl2010  and the
2019 PRad data XiEtAl2019  are discrepant Be2022priv  (see also Je2022talk ).
Clearly, further investigations are indicated.

##  IV Conclusions

In view of interesting possibilities with muon physics, one might ask if one
should add a muonic beam to TRIUMF/ARIEL. One idea for low-cost muon sources
can be mentioned, being based on an 8 GeV proton beam on a current-carrying
target followed by a lithium lens and a quadrupole decay channel, where the
decaying ions generate muons BaMo2011 .

It is highly unlikely that muonic hydrogen theory, and Lamb shift theory,
could provide explanations for the proton radius puzzle, since they are well
under control. From the experimental side, the situation regarding the proton
radius may be less clear than commonly thought. Electron versus muon
scattering experiments could shed light. A confirmation of the 1969 experiment
CaEtAl1969prl1 ; CaEtAl1969prl2 , which could point to a small electron-muon
nonuniversality, would be very desirable. A comparison of electron and muon
scattering on the same apparatus would shed light on the issue MUSE . Finally,
let us remember that an even more striking $7\sigma$ disagreement persists for
the deuteron radius, when comparing the results from electronic and muonic
bound systems PoEtAl2016 .

## Acknowledgments

Insightful conversations with J. C. Bernauer are gratefully acknowledged. This
research was supported by NSF grant PHY–2110294.

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# Variational properties of the abstract subdifferential operator

R. Díaz Millán Deakin University, 75 Pigdons Rd, Waurn Pond, VIC 3216 (Email:
<EMAIL_ADDRESS>N. Sukhorukova Swinburne University of Technology,
PO Box 218, Hawthorn, VIC 3122 (Email<EMAIL_ADDRESS>J. Ugon
Deakin University, 75 Pigdons Rd, Waurn Pond, VIC 3216 (Email:
<EMAIL_ADDRESS>
###### Abstract

Abstract convexity generalises classical convexity by considering the suprema
of functions taken from an arbitrarily defined set of functions. These are
called the abstract linear (abstract affine) functions. The purpose of this
paper is to study the abstract subdifferential. We obtain a number of results
on the calculus of this subdifferential: summation and composition rules, and
prove that under some reasonable conditions the subdifferential is a maximal
abstract monotone operator. Another contribution of this paper is a
counterexample that demonstrates that the separation theorem between two
abstract convex sets is generally not true. The lack of the extension of
separation results to the case of abstract convexity is one of the obstacles
in the development of abstract convexity based numerical methods.

Keywords: abstract convex functions, abstract subdifferential, abstract
monotonicity

MSC2020: 52A01, 90C30, 47N10, 49J52, 49J53

##  1 Introduction

Given a set $L$ of functions (also known as elementary functions), a function
is $L$-convex, or _abstract convex_ when it can be expressed as the supremum
of functions from $L$. Abstract convexity (also known as convexity without
linearity) has been extensively studied since the 1970s, and is the subject of
at least three monographs [20, 21, 22], and many papers [1, 2, 5, 6, 8, 15]
just to name a few. Many notions and results from classical convexity have
been generalised, including duality [15], the conjugate function [21], the
subdifferential [21] and normal cones [14]. The aim of this paper is to
investigate abstract subdifferential calculus, which will be a significant
step towards the development of abstract convexity based numerical methods.

In convex analysis the subdifferential plays an essential role for developing
algorithms. A major challenge in generalising the properties of the convex
subdifferential to the abstract convex subdifferential is that most proofs
rely on Minkowski’s theorem on the separation of sets. However, such a theorem
is not known in abstract convex analysis, and in fact, we show in this paper
that its generalisation is not true in general. More specifically, there might
not exist a function $l\in L$ separating two $L$-convex sets. Consequently, we
prove the results in this paper without resorting to such separation results.

Another challenge in the application of abstract convexity to practical
problems is the choice of an efficient set of elementary functions. In some
cases, for example uniform approximation problems, the choice is
straightforward, but this situation is rather exceptional. For some classes of
functions, such as quasiconvex functions, some elementary sets of functions
are known [21, 7] but counter-examples presented in [4] and in this paper show
that they do not always enable the generalisation of results such as the
maximality of the subdifferential operator, or the maximum rule.

The study of the abstract subdifferential is the subject of a few papers, and
several results, such as the sum rule or the maximal monotonicity of the
subdifferential operator [14, 6, 9] were generalised under some assumptions.
Nearly all of these papers make the assumption that the set $L$ is closed
under the sum (i.e., $L$ is additive). This is a strong assumption, which
excludes many families of functions of practical interest, such as quasilinear
and quasiconvex functions. It is also generally assumed that $0\in L$, which
is a more reasonable assumption, as this it is required to verify optimality
conditions. Finally several papers require that abstract convex functions are
additive, that is, they only consider functions $f$ such that
$f(x)+f(y)=f(x+y)$, also adding further hypotheses on the domain $X$ of these
functions. In most of the present paper we make no such assumption on $X$ and
$L$. In fact we derive a number of subdifferentiable calculus rules under very
general conditions, in such a way that these rules generalise fairly precisely
subdifferential calculus, both in the convex case and the known abstract
convex cases. Only in the last part of the paper, when we investigate the
maximal monotonicity of the subdifferential operator, do we introduce a few
assumptions on the set $X$.

The paper is divided into three parts. First, in Section 2 we recall some
important results and definitions of abstract convexity and clarify the
notations used throughout the paper. Then, in Section 3, we generalise several
well-known properties, including the sum rule on the subdifferential of the
sum of abstract convex functions and the composition rule. Finally, in Section
4, we prove that under suitable conditions the abstract subdifferential is a
maximal abstract monotone operator. Section 5 contains the conclusions and
open problems.

##  2 Preliminaries

We start by recalling the main definitions and properties of abstract
convexity. We will use the notations introduced by [21].

###  2.1 Definitions

Let $X$ be a set and $\mathcal{F}(X)$ be the set of all real-valued functions
with domain on $X$ as well as the function uniformly equal to $-\infty$:

$\mathcal{F}(X)=\\{f:X\to\mathds{R}\cup\\{+\infty\\}\\}\cup\\{-\infty:X\to\mathds{R},-\infty(x)=-\infty\\}.$

Let $H\subset\mathcal{F}(X)$ be a subset of finite functions.

######  Definition 1 (Abstract Convexity [21]).

A function $f$ is said to be _$H$ -convex_ if there exists a set $U\subset H$
such that for any $x\in X$, $f(x)=\sup_{u\in U}u(x)$.

######  Definition 2 (Support Set [21]).

The _$H$ -support set_ of a function $f\in\mathcal{F}(X)$ is defined as:

$\operatorname{supp}(f,H)=\\{u\in H:u\leq f\\}$

where $u\leq f$ means that $u(x)\leq f(x),\forall x\in X$.

The $H$-support sets of $H$-convex functions are called _$(H,X)$ -convex
sets_, or _$H$ -convex sets_ when the context is clear. Note that the
$H$-support set of every function is $H$-convex, and there is a one-to-one
mapping between $H$-convex functions and $H$-convex sets.

The intersections of $H$-convex sets is $H$-convex:

######  Proposition 1 ([8]).

Let $A_{i}:i\in I$ be a family of $H$-convex sets. Then $\cap_{i\in I}A_{i}$
is also $H$-convex.

For a set $L\subset\mathcal{F}(X)$, we denote by $H_{L}$ its vertical closure:
$H_{L}=\\{h_{lc}=l-c:l\in L,c\in\mathds{R}\\}$. The functions in $H_{L}$ are
called _abstract affine functions._ The set $L$ is called a set of _abstract
linear functions_ if $h_{l,c}\notin L$ for any $l\in L$ and $c\neq 0$.

######  Definition 3 (Abstract Convex Hull of a function [21]).

Let $f:X\to\mathds{R}$. The function $\operatorname{co}_{H}f$ defined by

$\operatorname{co}_{H}f(x)\coloneqq\sup\\{h(x):h\in\operatorname{supp}(f,H)\\}$

is called the _$H$ -convex hull_ of the function $f$.

It is easy to see that $\operatorname{co}_{H}f$ is the $H$-convex lower
envelope of the function $f$, that is, the supremum of all $H$-convex
functions that lie below $f$. In particular $f$ is $H$-convex if and only if
$f=\operatorname{co}_{H}f$. For a given set $L$, we will denote by $L_{x}$ the
set of abstract affine functions that all vanish at $x$:

$L_{x}=\\{l_{x}\coloneqq l-l(x):l\in L\\}.$ (1)

The set $L_{x}$ is a set of abstract linear functions with the same affine
closure as $L$, and its relevance will be clear when we discuss the abstract
subdifferential.

######  Definition 4 (Abstract Convex Hull of a set [21]).

The intersection of all $H$-convex sets containing a set $C\subset H$ is
called the _abstract convex hull_ of $C$ and denoted $\operatorname{co}_{H}C$.

######  Proposition 2 ([21, proposition 1.1 and Corollary 1.1]).

The $H$-convex hull of a set $C$ is $H$-convex. More specifically, it is the
smallest $H$-convex set containing $C$, and
$\operatorname{co}_{H}C=\operatorname{supp}(\sup_{l\in C}l,H)$.

Separation results are essential in convex analysis. The following proposition
is straightforward (separation of a point from a set).

######  Proposition 3 (Separating function [21]).

Given an $H$-convex set $U$ and a point $l\in H\setminus U$ there exists $x\in
X$: $l(x)>u(x)$ for any $u\in U$.

The separation of two disjoint $H$-convex sets does not generalise to the case
of abstract convexity. The following example illustrates that in general it is
not always possible to separate two disjoint $H$-convex sets by an $H$-linear
function.

######  Example 1.

Let $X=\mathds{R}$ and $H=\\{-\lvert x-1\rvert+2,-\lvert x+1\rvert+2,-\lvert
x\rvert+2,0\\}$. Consider the $H$-convex sets $A=\\{-\lvert
x-1\rvert+2,-\lvert x+1\rvert+2\\}$ and $B=\\{-\lvert x\rvert+2,0\\}$. Refer
to Figure 1.

It is easy to see that $A$ and $B$ are both $H$-convex, and disjoint. Indeed,
let

$f_{A}(x)=\sup_{u\in A}u(x).$

Then

$\\{x:-\lvert x\rvert+2>u(x)~{}\forall u\in A\\}=\\{x:\lvert
x\rvert+2>f_{A}(x)\\}=\left(-\frac{1}{2},\frac{1}{2}\right).$

Therefore $-\lvert x\rvert+2$ is not in
$\operatorname{supp}(f_{A},H)=\operatorname{co}_{H}A$. Likewise,

$\\{x:0>u(x)~{}\forall u\in A\\}=(-\infty,-3)\cup(3,+\infty).$

Therefore $0\notin\operatorname{supp}(f_{A},H)$. Yet, since
$[(-\infty,-3)\cup(3,+\infty)]\cap\left(-\frac{1}{2},\frac{1}{2}\right)=\emptyset$,
there is no point $x\in X$ that separates all functions in $B$ from $A$.

On the other hand, let $f_{B}(x)=\sup_{v\in B}v(x)$ then

$\\{-\lvert x\pm 1\rvert+2>v(x)~{}\forall v\in B\\}=\\{-\lvert x\pm
1\rvert+2>f_{B}(x)\\}=\left(\mp 3,\pm\frac{1}{2}\right),$

which shows that
$A\cap\operatorname{supp}(f_{B},H)=\operatorname{co}_{H}B=\emptyset$. However,
since

$\left(-3,-\frac{1}{2}\right)\cap\left(\frac{1}{2},3\right)=\emptyset,$

there is no point $x\in X$ that separates all functions in $A$ from $B$.

$-3$$-2$$-\frac{1}{2}$$\frac{1}{2}$$2$$3$$-3$$-2$$-1$$1$$2$$f_{A}$$A$$B$ (a)
Neither functions from $B$ are in the support set of the function $f_{A}$, and
so $A$ is $H$-convex. Yet, there is no point $x\in\mathds{R}$ at which both
functions from $B$ are above $f_{A}$. Therefore the set $B$ cannot be
separated from the set $A$.

$-3$$-2$$-\frac{1}{2}$$\frac{1}{2}$$2$$3$$-3$$-2$$-1$$1$$2$$f_{B}$$A$$B$ (b)
Neither functions from $A$ are in the support set of the function $f_{B}$, and
so $B$ is $H$-convex. Yet, there is no point $x\in\mathds{R}$ at which both
functions from $A$ are above $f_{B}$. Therefore the set $A$ cannot be
separated from the set $B$.

Figure 1: The sets $A$ and $B$ are both $H$-convex and disjoint, but they
cannot be separated by a point $x$.

######  Definition 5 (Abstract subdifferential [21]).

The $L$-subdifferential of a function at a point $x\in X$ is the set

$\partial_{L}f(x)=\\{l\in L:f(y)\geq f(x)+l(y)-l(x),\forall y\in X\\}.$

We recall that the sublevel sets and the epigraph of a function $f$ are
defined as follows:

$\displaystyle S_{c}(f)$ $\displaystyle=\\{x\in X:f(x)\leq c\\}$
$\displaystyle\operatorname{epi}(f)$ $\displaystyle=\\{(x,c)\in
X\times\mathds{R}:f(x)\leq c\\}$

When the domain of $f$ is ambiguous we denote the epigraph of $f$ over the
domain $X$ as $\operatorname{epi}_{X}f$.

Note that two functions are equal if and only if their epigraphs are equal.

######  Proposition 4.

A function $f$ is $(L,X)$-convex if and only if its epigraph is
$(X\times\mathds{R},L\times\\{-1\\})$-convex.

###### Proof.

Let $f$ be any function and $u\in\operatorname{supp}(f,L)$. Then,

$\displaystyle u\in\operatorname{supp}(f,L)$ $\displaystyle\iff$
$\displaystyle\forall y\in X,u(y)\leq f(y)$ $\displaystyle\iff$
$\displaystyle\forall y\in X,c\geq f(y),u(y)\leq c$ $\displaystyle\iff$
$\displaystyle\sup_{(y,c)\in\operatorname{epi}f}u(y)-c\leq 0$
$\displaystyle\iff$
$\displaystyle\forall(y,c)\in\big{(}\operatorname{co}_{(X\times\mathds{R}),(L\times\\{-1\\})}\operatorname{epi}f),u(y)-c\leq
0$

The function $f$ is $(L,X)$-convex if and only if for all $x\in X$,
$d\in\mathds{R}$ such that $d<f(x)$, there exists
$u\in\operatorname{supp}(f,L)$ such that $u(x)>d$. That is, for all
$(x,d)\notin\operatorname{epi}f$,
$(x,d)\notin\operatorname{co}_{(X\times\mathds{R}),(L\times\\{-1\\})}\operatorname{epi}f$.
This proves the result. ∎

######  Corollary 1.

For any function $f:X\to\mathds{R}$,
$\operatorname{co}_{(X\times\mathds{R},L\times\\{-1\\})}\operatorname{epi}f=\operatorname{epi}\operatorname{co}_{L}f$.

######  Definition 6 (Abstract Normal Set [14]).

The _abstract normal set_ at $x$ in $C$, with respect to the set of functions
$L$, denoted by $N_{L}(x,C)$ is defined as:

$N_{L}(x,C):=\\{l\in L:l(y)-l(x)\leq 0,\forall y\in C\\}.$

By convention we say that $N_{L}(x,C)=\emptyset$ when $x\notin C$.

Considering $(x,\lambda)\in X\times\mathds{R}$, $C\subset X\times\mathds{R}$
note that $N_{L\times\mathds{R}}((x,\lambda),C)=\\{(l,\alpha)\in
L\times\mathds{R}:l(y)-l(x)+\alpha(\gamma-\lambda)\leq 0,\forall(y,\gamma)\in
C\\}$.

######  Proposition 5.

Suppose that the function $f:X\to\mathds{R}$ is $L$-convex. Then we have the
following:

  1. 1.

For any $c\in\mathds{R}$, $S_{c}(f)$ is $(X,L)$-convex [15, 16];

  2. 2.

For $C\subset X$, $N_{L_{x}}(x,C)=\\{l-l(x):l\in N_{L}(x,C)\\}$ is
$(L_{x},X)$-convex, where $L_{x}\coloneqq\\{u-u(x):u\in L\\}$.

###### Proof.

  1. 1.

Define $g(l)=\sup_{x\in S_{c}(f)}l(x)$, and let $Y=\operatorname{supp}(g,L)$.
It is clear from the definition that $S_{c}(f)\subset Y$. Let $y\in Y$. Then,
$l(y)\leq g(l)$ for any $l\in L$, and in particular for any
$l\in\operatorname{supp}(f,L)$. In other words,

$l(y)\leq\sup_{x\in S_{c}(f)}l(x)\leq\sup_{x\in S_{c}(f)}f(x)\leq c$

Since this is true of any $l\in\operatorname{supp}(f,L)$, it is true of the
supremum, and so $f(y)\leq c$, which implies that $y\in S_{c}(f)$. Therefore
$Y=S_{c}(f)$, which is an $(X,L)$-convex set.

  2. 2.

For $y\in C$, define $N_{L}(x,y)=\\{l\in L:l(y)-l(x)\leq 0\\}=\\{l\in
L_{x}:l(y)\leq 0$. Note that $N_{L}(x,y)$ is the sublevel set of the function
$y:L_{x}\to\mathds{R}$ at the level 0, and therefore it is a $L_{x}$-convex
set.

Since

$\\{l-l(x):l\in N_{L}(x,C)\\}=\bigcap_{y\in C}N_{L}(x,y)$

we can conclude that $N_{L}(x,C)$ is a $L_{x}$-convex set, by Proposition 1.∎

Note that
$N_{L\times\\{-1\\}}((x,f(x)),\operatorname{epi}(f))=\partial_{L}f(x)\times\\{-1\\}$.

######  Proposition 6.

$\partial_{L}f(x)=\\{l\in L:(l,-1)\in
N_{L\times\mathds{R}}\left((x,f(x)),\operatorname{epi}(f)\right)\\}$.

###### Proof.

Consider $l\in\partial_{L}f(x)$, then for all $y\in X$, $f(y)\geq
f(x)+l(y)-l(x)$ implying that $l(y)-l(x)+(-1)(f(y)-f(x))\leq 0$, then
$(l,-1)\in N_{L}((x,f(x)),\operatorname{epi}(f))$.

On the other hand, suppose that $(l,-1)\in
N_{L}((x,f(x)),\operatorname{epi}(f))$, then for all
$(y,\lambda)\in\operatorname{epi}(f)$ which means that $\lambda\geq f(y)$, we
have $l(y)-l(x)+(-1)(\lambda-f(x))\leq 0$. In particular, since
$(y,f(y))\in\operatorname{epi}(f)$, then $l(y)-l(x)+(-1)(f(y)-f(x))\leq 0$
implying that $l\in\partial_{L}f(x)$. ∎

###  2.2 Abstract Conjugate

Let us recall a few results on the abstract conjugate.

######  Definition 7 (Abstract conjugate [21]).

The _abstract conjugate_ of the function $f:X\rightarrow\mathds{R}$, denoted
by $f^{*}:L\rightarrow\mathds{R}$ is defined by

$f^{*}(u):=\sup_{x\in X}\\{u(x)-f(x)\\}.$

Since for all $x\in X$, $f^{*}(u)\geq u(x)-f(x)$ we obtain the inequality

$f^{*}(u)+f(x)-u(x)\geq 0.$ (2)

######  Proposition 7 ([21, Proposition 7.7]).

Given $u\in L$ and $x\in X$, $f^{*}(u)+f(x)=u(x)$ if and only if
$u\in\partial_{L}f(x)$.

######  Proposition 8 ([14]).

Let $L$ be a family of functions, and $f$ be any function. We have
$\operatorname{epi}_{L}f^{*}=\operatorname{supp}(f,H_{L})$.

##  3 Abstract Subdifferential Calculus

In this section we develop a calculus for the abstract subdifferential, in
particular, the sum and composition rules. We start from the recalling the
following proposition from [8].

######  Theorem 1.

Let $L_{1}$ and $L_{2}$ families of abstract linear functions such that
$L_{1}\subset L_{2}$. Then for any $x\in X$,

$\partial_{L_{2}}f(x)\cap L_{1}=\partial_{L_{1}}f(x)$

###### Proof.

Suppose that $l\in L_{1}\cap\partial_{L_{2}}\partial f(x)$. Then for any $y\in
X$, $f(y)\geq f(x)-l(x)+l(y)$ and therefore $l\in\partial_{L_{1}}f(x)$.

If $l\in\partial_{L_{1}}f(x)$, it follows that $l\in L_{1}$ and that
$l\in\partial_{L_{2}}f(x)$. ∎

######  Lemma 1.

Let $f$ be a $L$-convex function, and let $L_{x}$ be defined as above. Then,
$\partial_{L_{x}}f(x)$ is a $L_{x}$-convex set.

###### Proof.

It suffices to prove that
$\operatorname{co}_{L_{x}}\partial_{L_{x}}f(x)\subset\partial_{L_{x}}f(x)$,
since the reverse inclusion is clear. Let $h=\sup_{l\in\partial_{L_{x}}f(x)}l$
and $u\in\operatorname{co}_{L_{x}}\partial_{L_{x}}f(x)$. Since
$\partial_{L_{x}}f(x)\subset L_{x}$, the function $h$ is $L_{x}$-convex, and
by definition of $\operatorname{co}_{L_{x}}$, the function $u$ is in the
$L_{x}$-support set of $h$. Then for any $y\in X$,

$\displaystyle u(y)-u(x)$ $\displaystyle\leq
h(y)=\sup_{l\in\partial_{L_{x}}f(x)}l(y)$
$\displaystyle\leq\sup_{l\in\partial_{L_{x}}f(x)}f(y)-f(x)+l(x)$
$\displaystyle=f(y)-f(x)$

from which we conclude that $u\in\partial_{L_{x}}f(x)$. ∎

######  Remark 1.

$\partial_{L}f(x)=\\{l\in L:l-l(x)\in\partial_{L_{x}}f(x)\\}$. Indeed,

$\displaystyle\partial_{L}f(x)$ $\displaystyle=\\{l\in L:\forall y\in
X,f(y)\geq f(x)+l(y)-l(x)\\}$ $\displaystyle=\\{l\in L:\forall y\in X,f(y)\geq
f(x)+l_{x}(y)-l_{x}(x)\\}$ $\displaystyle=\\{l\in
L:l-l(x)\in\partial_{L_{x}}f(x)\\}$

######  Remark 2.

Let $f$ be a function. We can develop simple rules to calculate the
$L$-subdifferential of the function obtained from $f$ through horizontal or
vertical shift:

Vertical shift:

For a function $u$, the $L-u$ subdifferential of the function $f_{u}=f-u$ at a
point $x\in X$ is $\partial_{L-u}f_{u}(x)=\partial_{L}f(x)-u$. Furthermore
$f_{u}$ is $(L-u)$-convex if and only if $f$ is $L$-convex.

Indeed,

$\displaystyle\partial_{L-u}f_{u}(x)$ $\displaystyle=\\{l_{u}\in L-u:\forall
z\in Xf_{u}(z)\geq f_{u}(x)+l_{u}(z)-l_{u}(x)\\}$
$\displaystyle=\\{\begin{multlined}l-u:\forall z\in X,l\in L,\\\ f(z)-u(z)\geq
f(x)-u(x)+l(z)-u(z)-l(x)+u(x)\\}\end{multlined}l-u:\forall z\in X,l\in L,\\\
f(z)-u(z)\geq f(x)-u(x)+l(z)-u(z)-l(x)+u(x)\\}$
$\displaystyle=\\{l-u:l\in\partial_{L}f(x)\\}$

Subdifferential vanishing at a point $y$:

For $y\in X$, the $L_{y}$-subdifferential of the function $f$ at a point $x\in
X$ is $\partial_{L_{y}}f(x)=\\{u-u(y):u\in\partial_{L}f(x)\\}$.

Horizontal shift:

For $y\in X$, define $L^{y}=\\{x\mapsto u(x+y):u\in L\\}$ and $X^{y}=X-y$. At
$x\in X^{y}$, the $L^{y}$ subdifferential of the function $f^{y}(x)=f(x+y)$ is
$\partial_{L^{y}}f(x)=\\{x\mapsto u(x+y):u\in\partial_{L}f(x+y)\\}$.
Furthermore $f^{y}$ is $(L^{y},X^{y})$-convex if and only if $f$ is
$(L,X)$-convex.

Indeed,

$\displaystyle\partial_{L_{y}}f(x)$ $\displaystyle=\\{l\in L_{y}:\forall z\in
X:f(z)\geq f(x)+l(z)-l(x)\\}$ $\displaystyle=\\{\begin{multlined}l-l(y):l\in
L,\forall z\in X:\\\ f(z)\geq
f(x)+l(z)-l(y)-l(x)+l(y)\\}\end{multlined}l-l(y):l\in L,\forall z\in X:\\\
f(z)\geq f(x)+l(z)-l(y)-l(x)+l(y)\\}$
$\displaystyle=\\{l-l(y):l\in\partial_{L}f(x)\\}$

Furthermore, it is easy to see that, for any ${y}\in X$ and $x\in X^{y}$,
$f_{y}(x)\coloneqq f(x+{y})$ is $L_{y}$-convex, where
$L_{y}=\\{u(\cdot+{y}):u\in L\\}$, and that
$\partial_{L_{y}}f_{y}(x)=\\{u(\cdot+{y}):u\in\partial_{L}f(x+y)\\}$. In fact,

$\displaystyle\partial_{L_{y}}f_{y}(x)$ $\displaystyle=$ $\displaystyle\\{u\in
L_{y}:f_{y}(z)\geq f_{y}(x)+u(y)-u(x),\forall z\in X^{y}\\}$ $\displaystyle=$
$\displaystyle\\{\begin{multlined}u(\cdot+y),u\in L:f(z+y)\geq
f(x+y)+u(z+y)-u(x+y),\\\ \forall z\in X^{y}\\}\end{multlined}u(\cdot+y),u\in
L:f(z+y)\geq f(x+y)+u(z+y)-u(x+y),\\\ \forall z\in X^{y}\\}$ $\displaystyle=$
$\displaystyle\\{u(\cdot+y),u\in L:f(t)\geq f(x+y)+u(t)-u(x+y),\forall t\in
X\\}$ $\displaystyle=$ $\displaystyle\\{u(\cdot+y):u\in\partial_{L}f(x+y)\\}$

######  Proposition 9.

Let $G\subset L$ be a finite subset of $L$ and let $f=\max_{g\in G}g$. Define
$A(x)=\\{g\in G:g(x)=f(x)\\}$. Then we have:

$\partial_{L_{x}}f(x)\supset\operatorname{co}_{L_{x}}A(x).$ (4)

###### Proof.

Let $h=\sup_{g\in A(x)}g$, and let
$U=\operatorname{supp}(h,L_{x})=\operatorname{co}_{L_{x}}A(x)$. By definition
of $L_{x}$ (see Formula (1)), we know that $h(x)=0$. Let $l\in\mathcal{F}(X)$
be such that $l_{x}=l-l(x)\in U$. Then, we have:

$\displaystyle l(y)-l(x)$ $\displaystyle\leq h(y)$
$\displaystyle\leq\sup_{g\in A}g(y)-g(x)$ $\displaystyle=\sup_{g\in
A}g(y)-f(x)$ $\displaystyle f(x)+l(y)-l(x)$ $\displaystyle=\sup_{g\in
A}g(y)=f(y)$

And therefore $l\in\partial_{\mathcal{F}(X)}f(x)$. In particular,
$l_{x}\in\partial_{\mathcal{F}(X)}f(x)\cap L_{x}=\partial_{L_{x}}f(x)$. ∎

The reverse inclusion in Proposition 9 is generally not true, as shown by the
following example.

######  Example 2.

Let $L$ be the set of all monotonous functions over $\mathds{R}$ vanishing at
0. Consider the $L$-convex function $\max(g_{1}(x),g_{2}(x))$, where
$g_{1}(x)=x$ and $g_{2}(x)=-x$ at $x=1$. At this point, $A(1)=\\{g_{1}\\}$.
The function $u(x)=\max(0,x)-1$ is an $L_{x}$-subgradient of $f$ at $x=1$ (See
Figure 2), since for any $y\in\mathds{R}$, we have $f(y)=\lvert y\rvert\geq
1+(\max(0,y)-1)-0=f(x)+u(y)-u(x)$. However, $u$ does not lie in
$\operatorname{co}_{L_{x}}(t\mapsto g_{1}(t)-t)$, since for $y=-1$, we have
$g_{1}(y)-1=-1-1<1-1-1+0=g_{1}(x)+u(-1)-u(1)$. Therefore
$\max(0,x)\in\partial_{L_{1}}f(1)\setminus\partial_{L_{1}}g_{1}(1)$.

$1$$-2$$-1$$1$$2$$\max(g_{1},g_{2})$$u$ Figure 2: The function $u$ is a
$L_{1}$-subgradient of the function $|x|$ at $x=1$.

######  Proposition 10 (Composition Rule).

Let $f$ be an $(L,X)$-convex function, and $u:Y\to X$. Then

$(\partial_{L}f(x))\circ u\subset\partial_{L\circ u}(f\circ u)(x),$

where $L\circ u=\\{l\circ u:l\in L\\}$.

Furthermore, if $u$ is onto the domain of $f$, then we have equality:

$(\partial_{L}f(x))\circ u=\partial_{L\circ u}(f\circ u)(x),$

###### Proof.

Let $w\in\partial_{L}f(u(x))\circ u$, that is, there exists
$v\in\partial_{L}f(u(x))$ such that $w=v\circ u$. Then, we have, for any $y\in
X$:

$f(y)\geq f(u(x))+v(u(y))-v(u(x)).$

This is true, in particular, when $y\in\operatorname{dom}u$. In other words,
for any $z\in Y$,

$f(u(z))\geq f(u(x))+v(u(y))-v(u(x))$

and therefore $w\in\partial_{L\circ u}f\circ u(x)$.

When the image of $u$ $\operatorname{Im}u=\operatorname{dom}f$, and
$w\in\partial_{L\circ u}f\circ u(x)$, letting $w=v\circ u$, then for any
$y\in\operatorname{dom}f$, there exists $z\in Y$ such that $y=u(z)$. Then:

$f(y)=f(u(z))\geq f(u(x))+v(u(y))-v(u(x)).$

So, $v\in\partial_{L}f(u(x))$, which proves the reverse inclusion. ∎

We now turn our attention to the sum of abstract convex functions. We start
from recalling the definition of infimum convolution.

######  Definition 8 (Infimum Convolution[14]).

Let $f:X_{1}\to\mathds{R}$ and $g:X_{2}\to\mathds{R}$ be real-valued
functions. The infimal convolution $f\oplus g:(X_{1}+X_{2})\to\mathds{R}$ is
defined as:

$(f\oplus g)(x)=\inf_{\begin{subarray}{c}x_{1}+x_{2}=x\\\ x_{1}\in X_{2}\\\
x_{2}\in X_{2}\end{subarray}}\big{(}f(x_{1})+g(x_{2})\big{)}.$

The convolution is said to be _exact_ when the infimum is attained for every
$x\in X$.

Note that unlike in most definitions we do not assume $f$ and $g$ to be
defined on the same domain. This is so we can apply the infimal convolution to
the conjugates of $L_{1}$\- and $L_{2}$-convex functions.

We now give some results on the sums of convex sets and convex functions. We
let $f_{1}$ and $f_{2}$ be two functions defined on $X$, and $L_{1}$ and
$L_{2}$ any two families of functions on $X$.

######  Proposition 11.

If $f_{1}$ is $L_{1}$-convex and $f_{2}$ is $L_{2}$-convex, then $f_{1}+f_{2}$
is $L_{1}+L_{2}$-convex.

###### Proof.

Let $U_{i}=\operatorname{supp}(f_{i},L_{i})$ for $i\in\\{1,2\\}$. We have that

$\displaystyle(f_{1}+f_{2})(x)$ $\displaystyle=\sup\\{u_{1}(x):x\in
U_{1}\\}+\sup\\{u_{2}(x):x\in U_{2}\\}$
$\displaystyle=\sup\\{(u_{1}(x)+u_{2}(x):u_{1}\in U_{1},u_{2}\in
U_{2}\\}.\qed$

######  Proposition 12.

$\operatorname{supp}(f_{1}+f_{2},L_{1}+L_{2})=\operatorname{co}_{L_{1}+L_{2}}\Big{(}\operatorname{supp}(f_{1},L_{1})+\operatorname{supp}(f_{2},L_{2})\Big{)}$

.

###### Proof.

We have

$\displaystyle\operatorname{co}_{L_{1}+L_{2}}(f_{1}+f_{2})(x)$
$\displaystyle=\operatorname{co}_{L_{1}+L_{2}}\left(\sup_{l_{1}\in\operatorname{supp}(f_{1},L_{1})}l_{1}(x)+\sup_{l_{2}\in\operatorname{supp}(f_{2},L_{2})}l_{2}(x)\right)$
$\displaystyle\sup_{l\in\operatorname{supp}(f_{1}+f_{2},L_{1}+L_{2})}l(x)$
$\displaystyle=\operatorname{co}_{L_{1}+L_{2}}\sup_{l_{1}+l_{2}\in\operatorname{supp}(f_{1},L_{1})+\operatorname{supp}(f_{2},L_{2})}l_{1}(x)+l_{2}(x)$
$\displaystyle=\sup_{l_{1}+l_{2}\in\operatorname{supp}(f_{1},L_{1})+\operatorname{supp}(f_{2},L_{2})}l_{1}(x)+l_{2}(x)$

Applying Proposition 2 gives the result. ∎

######  Proposition 13.

$\operatorname{epi}_{L_{1}+L_{2}}(f_{1}^{*}\oplus
f_{2}^{*})=\operatorname{epi}_{L_{1}}f_{1}^{*}+\operatorname{epi}_{L_{2}}f_{2}^{*}$

###### Proof.

We have:

$\displaystyle\operatorname{epi}_{L_{1}+L_{2}}(f_{1}^{*}\oplus f_{2}^{*})$
$\displaystyle=\\{(l,c):\inf_{l_{1}+l_{2}=l}\big{(}f_{1}^{*}(l_{1})+f_{2}^{*}(l_{2})\big{)}\leq
c\\}$ $\displaystyle\begin{aligned} =\\{(l,c)&:\exists(l_{1},c_{1})\in
L_{1}\times\mathds{R},l_{2}\in(L_{2}\times\mathds{R}):\\\
&f_{1}^{*}(l_{1})\leq c_{1},f_{2}^{*}(l_{2})\leq
c_{2},l_{1}+l_{2}=l,c_{1}+c_{2}=c\\}\end{aligned}$
$\displaystyle=\operatorname{epi}_{L_{1}}f_{1}^{*}+\operatorname{epi}_{L_{2}}f_{2}^{*}\qed$

######  Corollary 2.

$(f_{1}+f_{2})^{*}=\operatorname{co}_{L_{1}+L_{2}}(f_{1}^{*}\oplus f_{2}^{*})$

###### Proof.

By Propositions 8 and 13, we have that

$\operatorname{epi}_{L_{1}+L_{2}}(f_{1}^{*}\oplus
f_{2}^{*})=\operatorname{supp}(f_{1},H_{L_{1}})+\operatorname{supp}(f_{2},H_{L_{2}})$
(5)

Applying Proposition 12, this implies that

$\displaystyle\operatorname{co}_{H_{L_{1}}+H_{L_{2}}}\operatorname{epi}_{L_{1}+L_{2}}(f_{1}^{*}\oplus
f_{2}^{*})$
$\displaystyle=\operatorname{supp}(f_{1}+f_{2},H_{L_{1}}+H_{L_{2}})$
$\displaystyle=\operatorname{epi}(f_{1}+f_{2})^{*}$

The result then follows from Corollary 1. ∎

######  Corollary 3.

The following statements are equivalent:

  1. 1.

$\operatorname{supp}(f_{1}+f_{2},H_{L_{1}}+H_{L_{2}})=\operatorname{supp}(f_{1},H_{L_{1}})+\operatorname{supp}(f_{2},H_{L_{2}})$

  2. 2.

$(f_{1}^{*}\oplus f_{2}^{*})=(f_{1}+f_{2})^{*}$ with exact infimal
convolution.

  3. 3.

$\operatorname{epi}_{L_{1}}f_{1}^{*}+\operatorname{epi}_{L_{2}}f_{2}^{*}=\operatorname{epi}_{L_{1}+L_{2}}(f_{1}+f_{2})^{*}$.

###### Proof.

The equivalence of the first two statements derives directly from Equation
(5), since the $(H_{L_{1}}+H_{L_{2}})$-convexity of the left hand side is
equivalent to the convexity of the right hand side.

When this happens we have that for any $l\in L_{1}+L_{2}$,

$(l,(f_{1}^{*}\oplus
f_{2}^{*})(l))\in\operatorname{epi}_{L_{1}+L_{2}}(f_{1}^{*}\oplus
f_{2}^{*})=\operatorname{supp}(f_{1}+f_{2},H_{L_{1}}+H_{L_{2}}).$

Then there exist $(l_{1},c_{1})\in H_{L_{1}}$, $(l_{2},c_{2})\in H_{L_{2}}$,
such that $l_{1}+l_{2}=l$ and $c_{1}+c_{2}=(f_{1}^{*}\oplus f_{2}^{*})(l)$,
and

$(f_{1}^{*}\oplus f_{2}^{*})(l)\leq f_{1}^{*}(l_{1})+f_{2}^{*}(l_{2})\leq
c_{1}+c_{2}=(f_{1}^{*}\oplus f_{2}^{*})(l)$

The second and third statements are equivalent by Proposition 13. ∎

Note that the statements above can be interpreted as follows: the sum of
abstract convex sets (as the support sets of two abstract convex functions) is
abstract convex if and only if the infimal convolution of the two functions is
abstract convex.

Define the support function of the set $C\subset X$ as
$\sigma_{C}:L\rightarrow\mathds{R}$, $\sigma_{C}(l):=\sup_{x\in C}l(x)$ and
the indicator function of $C$ by $\delta_{C}(x)\coloneqq 0$ if $x\in C$ and
$\delta_{C}(x)\coloneqq+\infty$ if $x\notin C$. It is easy to see that
$\sigma_{C}=\delta_{C}^{*}$.

######  Corollary 4.

Let $C\subset X$ and $D\subset X$ be subsets of $X$. Then,

$\operatorname{epi}_{L_{1}+L_{2}}(\sigma_{C\cap
D})=\operatorname{co}_{(L_{1}+L_{2})\times\mathds{R}}\big{(}\operatorname{epi}_{L_{1}}\sigma_{C}+\operatorname{epi}_{L_{2}}\sigma_{D}\big{)}.$

###### Proof.

We have from Corollary 2

$\sigma_{C\cap
D}=(\delta_{C}+\delta_{D})^{*}=\operatorname{co}_{L_{1}+L_{2}}(\delta_{C}^{*}\oplus\delta_{D}^{*})$

Applying Corollary 1, this implies that

$\displaystyle\operatorname{epi}_{L_{1}+L_{2}}\sigma_{C\cap D}$
$\displaystyle=\operatorname{co}_{(L_{1}+L_{2})\times\mathds{R}}\operatorname{epi}_{L_{1}+L_{2}}(\delta_{C}^{*}\oplus\delta_{D}^{*})$
$\displaystyle=\operatorname{co}_{(L_{1}+L_{2})\times\mathds{R}}(\operatorname{epi}_{L_{1}}\delta_{C}^{*}+\operatorname{epi}_{L_{2}}\delta_{D}^{*})$

where the latter equality follows from Proposition 13. From there and the fact
that $\delta_{C}^{*}=\sigma_{C}$ and $\delta_{D}^{*}=\sigma_{D}$ we conclude
that

$\operatorname{epi}_{L_{1}+L_{2}}(\sigma_{C\cap
D})=\operatorname{co}_{(L_{1}+L_{2})\times\mathds{R}}(\operatorname{epi}_{L_{1}}\sigma_{C}+\operatorname{epi}_{L_{2}}\sigma_{D}).\qed$

Note that the definition of the support function implies $N_{L}(x,C)=\\{l\in
L:\sigma_{C}(l)=l(x)\\}$.

######  Proposition 14.

Consider $C,D\subset X$ such that $C$ is $(X,L_{1})$-convex and $D$ is
$(X,L_{2})$-convex and suppose that
$\operatorname{supp}(\delta_{C},H_{L_{1}})+\operatorname{supp}(\delta_{D},H_{L_{2}})=\operatorname{supp}(\delta_{C}+\delta_{D},H_{L_{1}+L_{2}})$
Then for all $x\in X$, $N_{L}(x,C\cap D)=N_{L_{1}}(x,C)+N_{L_{2}}(x,D)$.

###### Proof.

First, if $x\notin C\cap D$, that is, $x\notin C$ or $x\notin D$, then the
abstract normal set on both sides of the equality is the empty set.

Now consider $x\in C\cap D$. Take $l\in N_{L_{1}}(x,C)+N_{L_{2}}(x,D)$, then
there exists $l_{C}\in N_{L_{1}}(x,C)$ and $l_{D}\in N_{L_{2}}(x,D)$ such that
$l=l_{C}+l_{D}$. For all $y\in C$ $l_{C}(y)\leq l_{C}(x)$ and for all $z\in D$
$l_{D}(z)\leq l_{D}(x)$, then taking any point $w\in C\cap D$ we have, summing
the inequalities above, $l_{C}(w)+l_{D}(w)\leq l_{C}(x)+l_{D}(x)$. So, for all
$w\in C\cap D$, $l(w)\leq l(x)$, implying that $l\in N_{L}(x,C\cap D)$.

On the other hand, consider $l\in N_{L}(x,C\cap D)$. We have, by assumption
that
$\operatorname{supp}(\delta_{C},H_{L_{1}})+\operatorname{supp}(\delta_{D},H_{L_{2}})=\operatorname{supp}(\delta_{C}+\delta_{D},H_{L_{1}+L_{2}})$
and Corollary 3, that

$(\sigma_{C}\oplus\sigma_{D})(l)=(\delta_{C}+\delta_{D})^{*}(l)=\delta_{C\cap
D}^{*}(l)=\sigma_{C\cap D}(l)=l(x).$

and since the infimal convolution is exact, there exist $l_{1}\in L_{1}$ and
$l_{2}$ in $L_{2}$ with $l_{1}+l_{2}=l$ such that
$\sigma_{C}(l_{1})+\sigma_{D}(l_{2})=l(x)$.

Now, for all $z\in D$,

$0\geq
l_{1}(x)-\sigma_{C}(l_{1})=(l-l_{2})(x)+\sigma_{D}(l_{2})-l(x)=\sigma_{D}(l_{2})-l_{2}(x)\geq
l_{2}(z)-l_{2}(x).$

This implies that $l_{2}\in N_{L_{2}}(x,D)$. In the same way we can prove that
$l_{1}\in N_{L_{1}}(x,C)$. Then $N_{L}(x,C\cap D)\subseteq
N_{L_{1}}(x,C)+N_{L_{2}}(x,D)$. This proves the equality. ∎

######  Theorem 2 (Sum Rule).

Consider two families of functions $L_{1}\in\mathcal{F}(X)$ and
$L_{2}\in\mathcal{F}(X)$, and two functions $f_{1}:X\to\mathds{R}$ and
$f_{2}:X\to\mathds{R}$ be $L_{1}$-convex and $L_{2}$-convex respectively such
that $\operatorname{dom}f_{1}\cap\operatorname{dom}f_{2}\neq\emptyset$. Assume
that
$\operatorname{supp}(f_{1},H_{L_{1}})+\operatorname{supp}(f_{2},H_{L_{2}})=\operatorname{supp}(f_{1}+f_{2},H_{L_{1}}+H_{L_{2}})$
Let $f=f_{1}+f_{2}$ and $L=L_{1}+L_{2}$. Then for all
$x\in\operatorname{dom}f_{1}\cap\operatorname{dom}f_{2}$, we have the
following:

$\partial_{L_{1}+L_{2}}(f_{1}+f_{2})=\partial_{L_{1}}f_{1}(x)+\partial_{L_{2}}f_{2}(x).$

###### Proof.

Take $u_{1}\in\partial_{L_{1}}f_{1}(x)$ and
$u_{2}\in\partial_{L_{2}}f_{2}(x)$. Then, for any $y\in X$,

$\displaystyle f(y)$ $\displaystyle=f_{1}(y)+f_{2}(y)$ $\displaystyle\geq
f_{1}(x)+u_{1}(y)-u_{1}(x)+f_{2}(x)+u_{2}(y)-u_{2}(x)$
$\displaystyle=f(x)+(u_{1}+u_{2})(y)-(u_{1}+u_{2})(y)$

Therefore $u_{1}+u_{2}\in\partial_{L_{1}+L_{2}}(f_{1}+f_{2})(x)$.

For the reverse inclusion, take $u\in\partial_{L_{1}+L_{2}}f(x)$ and consider
the sets.

$\displaystyle\Omega_{1}$
$\displaystyle\coloneqq\\{(x,\lambda_{1},\lambda_{2})\in
X\times\mathds{R}\times\mathds{R}:\lambda_{1}\geq f_{1}(x)\\}$
$\displaystyle\Omega_{2}$
$\displaystyle\coloneqq\\{(x,\lambda_{1},\lambda_{2})\in
X\times\mathds{R}\times\mathds{R}:\lambda_{2}\geq f_{2}(x)\\}.$

For $u_{1}\in L_{1}$, $\sigma_{\Omega_{1}}(u_{1},-1,0)=\sup_{x\in
X,\lambda\geq f(x)}u_{1}(x)-\lambda=\sup_{x\in
X}u_{1}(x)-f_{1}(x)=f_{1}^{*}(u_{1})$. Similarly,
$\sigma_{\Omega_{2}}(v,0,-1)=f_{2}^{*}(v)$. Therefore,
$\sigma_{\Omega_{1}}(u,-1,0)+\sigma_{\Omega_{2}}(v,0,-1)=f_{1}^{*}(u)+f_{2}^{*}(v)$.

By assumption we have:
$\operatorname{supp}(f_{1},H_{L_{1}})+\operatorname{supp}(f_{2},H_{L_{2}})=\operatorname{supp}(f_{1}+f_{2},H_{L_{1}}+H_{L_{2}})$,
and, by Corollary 3, there exist $l_{1}\in L_{1}$ and $l_{2}\in L_{2}$ such
that
$(f_{1}+f_{2})^{*}(l_{1}+l_{2})=f_{1}^{*}(l_{1})+f_{2}^{*}(l_{2})=\sigma_{1}(l_{1},-1,0)+\sigma_{2}(l_{2},0,-1)$.

In particular, $\sigma_{1}\oplus\sigma_{2}$ is
$\big{(}(L_{1}+L_{2})\times(\\{-1\\}+\\{0\\})\times(\\{0\\}+\\{-1\\})\big{)}$-convex.
Applying Corollary 2 this means that
$(\delta_{\Omega_{1}}+\delta_{\Omega_{2}})^{*}=\delta^{*}_{\Omega_{1}}\oplus\delta^{*}_{\Omega_{2}}$.
From this we can apply Corollary3 to conclude that
$\operatorname{supp}(\delta_{\Omega_{1}},H_{L_{1}})+\operatorname{supp}(\delta_{\Omega_{2}},H_{L_{2}})=\operatorname{supp}(\delta_{\Omega_{1}}+\delta_{\Omega_{2}},H_{L_{1}}+H_{L_{2}})$.

Fixing $x\in\operatorname{dom}(f_{1})\cap\operatorname{dom}(f_{2})$ and
considering $\lambda=\max(f_{1}(x),f_{2}(x))$, then
$(x,\lambda,\lambda)\in\Omega_{1}\cap\Omega_{2}\neq\emptyset$. We claim that
$(u,-1,-1)\in
N_{L}\left((x,f_{1}(x),f_{2}(x)),\Omega_{1}\cap\Omega_{2}\right)$. Indeed for
all $(y,\lambda,\alpha)\in\Omega_{1}\cap\Omega_{2}$, since
$u\in\partial_{L}f(x)$, we have

$u(y)-u(x)+(-1)(\lambda-f_{1}(x))+(-1)(\alpha-f_{2}(x))\leq
f_{1}(y)-\lambda+f_{2}(y)-\alpha\leq 0.$

Invoking Proposition 14, we obtain that

$N_{L}\left((x,f_{1}(x),f_{2}(x)),\Omega_{1}\cap\Omega_{2}\right)=\\\
N_{L_{1}}\left((x,f_{1}(x),f_{2}(x)),\Omega_{1}\right)+N_{L_{2}}\left((x,f_{1}(x),f_{2}(x)),\Omega_{2}\right).$

Now, note that for all $(x,\lambda,\alpha)\in
N_{L}\left((x,f_{1}(x),f_{2}(x)),\Omega_{1}\right)$, we have $\alpha=0$, and
$(x,\lambda,\alpha)\in N_{L}\left((x,f_{1}(x),f_{2}(x)),\Omega_{2}\right)$
implies that $\lambda=0$. Then $(v,-1,-1)=(u_{1},\lambda,0)+(u_{2},0,\alpha)$,
and $u=u_{1}+u_{2}$ and $\lambda=\alpha=-1$. Using Proposition 6, we have that
$u_{1}\in\partial_{L}f_{1}(x)$ and $u_{2}\in\partial_{L}f_{2}(x)$ as desired.
∎

Proposition 2 a generalises the conditions obtained by [3] in the context of
the classical lower semi-continuous convex functions (recall that lower semi-
continuous convex functions are abstract convex with respect to linear
functions, and that abstract convex sets are closed convex sets [21] so that
the abstract convex hull of a set is its weak* closure.) They also generalise
the results by [14] in the case when the set $L$ is additive and $0\in L$.

##  4 The abstract subdifferential as an abstract monotone operator

This section is dedicated to the study of the abstract subdifferential of an
L-convex function. Following we return to the definition of abstract
monotonicity. From here onward we assume that the set $X$ is a Banach space.

######  Definition 9 (Abstract monotonicity [20, Eq. (1.1.8)]).

An operator $T:X\rightrightarrows L$ is said to be abstract monotone with
respect to $L$ if for any $x,y\in X$, $u\in T(x)$, $v\in T(y)$,
$u(x)-u(y)+v(y)-v(x)\geq 0$.

Furthermore $T$ is _maximal_ if it is maximal with respect to set inclusion:
if $(y,v)\in X\times L$, if $u(x)-u(y)+v(y)-v(x)\geq 0$ for all $(x,u)\in T$,
then $v\in T(y)$.

The inverse operator, denoted by $T^{-1}:L\rightrightarrows X$, is defined by
$T^{-1}(u):=\\{x\in X:u\in T(x)\\}$.

######  Proposition 15.

Consider two abstract monotone operators $T_{1}:X\to L_{1}$ and $T_{2}:X\to
L_{2}$ . Then

  1. 1.

$T_{1}^{-1}$ is $X$-abstract monotone; furthermore if $T_{1}$ is abstract
maximal monotone, then so is $T_{1}^{-1}$.

  2. 2.

For any $\lambda_{1}\geq 0,\lambda_{2}\geq 0$,
$\lambda_{1}T_{1}+\lambda_{2}T_{2}$ is $\lambda_{1}L_{1}+\lambda_{2}L_{2}$ is
abstract monotone.

###### Proof.

  1. 1.

Let $u,v\in L_{1}$ and $x,y\in X$ be such that $x\in T_{1}^{-1}(u)$ (that is,
$u\in T_{1}(x)$) and $y\in T_{1}^{-1}(v)$ (that is, $v\in T_{1}(y)$). Then it
is clear that $u(x)-u(y)+v(y)-v(x)\geq 0$, implying that $T_{1}^{-1}$ is
abstract monotone. Now consider a pair $(u,x)\in L_{1}\times X$ such that for
all $(v,y)\in L_{1}\times X$ such that $y\in T^{-1}(v)$,
$u(x)-u(y)+v(y)-v(x)\geq 0$. If $T_{1}$ is maximal, then we have that $u\in
T_{1}(x)$, that is, $x\in T_{1}^{-1}(u)$, proving the maximality of $T^{-1}$.

  2. 2.

Let $x,y\in X$ and $u\in(\lambda_{1}T_{1}+\lambda_{2}T_{2})(x)$ and
$v\in(\lambda_{1}T_{1}+\lambda_{2}T_{2})(v)$. Then there exist $u_{1}\in
T_{1}(x)$ and $u_{2}\in T_{2}(x)$ such that
$u=\lambda_{1}u_{1}+\lambda_{2}u_{2}$ and $v_{1}\in T_{1}(y)$ and $v_{2}\in
T_{2}(y)$ such that $v=\lambda_{1}v_{1}+\lambda_{2}v_{2}$. We have:

$u(x)-u(y)+v(y)-v(x)=\lambda_{1}(u_{1}(x)-u_{1}(y)+v_{1}(y)-v_{1}(x))\\\
+\lambda_{2}(u_{2}(x)-u_{2}(y)+v_{2}(y)-v_{2}(x))\geq 0.$

∎

######  Assumption 1.

Given a L-convex function $f:X\to\mathds{R}$, for each $a>0$ and any $x\in X$,
$\operatorname{supp}(f,H_{L})+\operatorname{supp}(\|\cdot-x\|,H_{\mathcal{L}})$
is $H_{L+\mathcal{L}}$-convex, and the function $f+\|\cdot-x\|$ is bounded
from below on the domain of $f$.

The next result is a generalisation of the Brønsted-Rockafellar theorem, which
is important to prove the main result of this section.

######  Theorem 3 (Brønsted-Rockafellar theorem extended).

Let $f$ be a proper lsc $L$-convex function, $y\in\operatorname{dom}f$
satisfying Assumption 1 and $v\in\operatorname{dom}f_{L}^{*}$, and let
$\lambda\geq 0$ and $\mu\geq 0$ be such that $f(y)+f^{*}_{L}(v)\leq
v(y)+\lambda\mu$. Then, there exists $z\in X$ and $w\in\partial_{L}f(z)$ such
that $\|z-y\|\leq\lambda$ and $w-v=\langle p,\cdot\rangle$, where $p\in X^{*}$
such that $\|p\|\leq\mu$.

###### Proof.

Set $\alpha=\lambda\mu$. We can assume that $\alpha>0$ as otherwise, by
Proposition 7, we have $v\in\partial_{L}f(y)$ and we are done. Define $h=f-v$.
It can be seen, from Remark 2 (1), that $h$ is $L-\\{v\\}$-convex, and for any
$x\in X$, $\partial_{L-\\{v\\}}h(x)=\partial_{L}f(x)-\\{v\\}$. For any $x\in
X$, according to (2) we have that

$h(x)=-(v(x)-f(x))\geq-f^{*}_{L}(v)\geq f(y)-v(y)-\lambda\mu=h(y)-\alpha.$

Hence, $h$ is bounded from below and $\alpha+\inf_{x\in X}h(x)\geq h(y)$.

Applying Ekeland’s principle (Theorem 1.1 in [12]) to $h$, we get the
existence of $z\in X$ such that $\|z-y\|\leq\lambda$ and
$z=\mathop{\mathrm{argmin}}(h+\frac{\alpha}{\lambda}\|\cdot-z\|)$. Define
$\tilde{h}:=h+\frac{\alpha}{\lambda}\|\cdot-z\|$.

By Proposition 11, $\tilde{h}$ is in $((L-v)+\mathcal{L})$-convex, where
$\mathcal{L}$ is the set of linear functions, and so
$0\in\partial_{L-\\{v\\}+\mathcal{L}}\tilde{h}(z)$. Since Assumption 1 holds,
we can apply Theorem 2 to find that

$0\in\partial_{L}f(z)-\\{v\\}+\mu\\{\langle p,\cdot\rangle:p\in B(0,1)\\}.$

Hence, there exists $w\in\partial_{L}f(z)$ such that $v-w\in\\{\langle
p,\cdot\rangle:\|p\|\leq\mu\\}$. ∎

######  Remark 3.

Note that $\\{v\\}\subseteq\\{v+\langle p,\cdot\rangle:\|p\|\leq\mu\\}\cap
L\neq\emptyset$. When equality occurs, this implies that $v=w$ in the theorem
above.

Our next result concerns the maximal monotonicity of the abstract
subdifferential operator. It is not hard to see that the $L$-subdifferential
is $L$-monotone [20, Proposition 1.9]. However, [4, example 3.1] showed that
it is not generally maximal $L$-monotone. Under which condition it has been
investigated in [10, 11, 18, 19, 17] among others. Here we show that under
Assumption 1 the result is also true. Our proof generalises the proof by [13].
We will need the following assumptions and the proposition below.

######  Example 3.

Let $X$ be a Banach space, and $f$ any convex function. Since the domain of
$g_{a}\coloneqq a\|\cdot-x\|$ is $X$ for any $x$ and any $a>0$, then
$\operatorname{ri}(\operatorname{dom}(f))\cap\operatorname{ri}(\operatorname{dom}(g_{a}))$
is nonempty, and it is known that this implies that
$\operatorname{epi}f^{*}+\operatorname{epi}g_{a}^{*}=\operatorname{epi}(f+g_{a})^{*}$
(see [5]). From this we infer that $f$ satisfies Assumption 1.

######  Proposition 16.

Let $f$ be a proper lsc $L$-convex function that satisfies Assumption 1 and

$u(x)\geq 0,\forall x\in\operatorname{dom}f,u\in\partial_{L}f(x).$ (6)

Then $0\in\partial_{L}f(0)$.

###### Proof.

For $a>0$, define $g_{a}\coloneqq a\|\cdot\|$.

If $p\in\partial_{\mathcal{L}}g_{a}(x)$ then, $\langle p,0-x\rangle\leq
a\|0\|-a\|x\|$, that is,

$p(x)\geq g_{a}(x).$ (7)

Define $f_{a}(x)\coloneqq f(x)+g_{a}(x)$. Then
$\partial_{L+\mathcal{L}}f_{a}(x)=\partial_{L}f(x)+\partial_{\mathcal{L}}g_{a}(x)$.
Let $p\in\partial_{L+\mathcal{L}}f_{a}(x)$. Then there exist
$p_{1}\in\partial_{L}f(x)$ and $p_{2}\in\partial_{\mathcal{L}}g_{a}(x)$ such
that $p=p_{1}+p_{2}$. Then, we have that

$p(x)=p_{1}(x)+p_{2}(x)\geq 0+g_{a}(x)=g_{a}(x).$ (8)

Let $x_{n}$ be a minimising sequence of $f_{a}$, that is
$f_{a}(x_{n})<f_{a}(x)+\varepsilon_{n}$, $\forall x\in X$ for some
$\varepsilon_{n}\to 0$. Equivalently
$0\in\partial_{L+\mathcal{L},\varepsilon_{n}}f_{a}(x_{n})$. Applying Theorem
3, we find that there exists $y_{n}\in B(x_{n},\sqrt{\varepsilon_{n}})$ and
$p_{n}\in\partial_{L+\mathcal{L}}f_{a}(y_{n})$ such that $\exists l\in
B(0,\sqrt{\varepsilon_{n}}):p_{n}-0=\langle l,\cdot\rangle$.

Therefore we have $g_{a}(y_{n})\to 0$, and by continuity of $g_{a}$, $x_{n}\to
0$. So, $0$ is the global minimiser of $f_{a}$.

In other words, $f_{a}(x)\geq f_{a}(0)$. Equivalently, $f(x)\geq
f(0)-g_{a}(x)=f(0)-\|x\|$. Since $a$ was arbitrary, we conclude that $f(x)\geq
f(0)$ and therefore $0\in\partial_{L}f(0)$. ∎

######  Theorem 4.

Let $f$ be an $L$-convex function satisfying Assumption 1. Then
$\partial_{L}f$ is a maximal $L$-monotone operator.

###### Proof.

First we show that $\partial_{L}f$ is $L$-monotone. Let $x,y\in X$, and
$u\in\partial_{L}f(x)$ and $v\in\partial_{L}f(y)$. Then by definition of the
abstract subdifferential,

$f(x)\geq f(y)+u(y)-u(x)\geq f(x)+v(x)-v(y)+u(y)-u(x).$

Therefore

$v(y)-v(x)+u(x)-u(y)\geq 0.$

Now suppose that $(y,v)\in X\times L$ is such that

$u(x)-u(y)+v(y)-v(x)\geq 0$ (9)

for all $x\in X$, $u\in\partial_{L}f(x)$.

Define $\bar{f}(x)\coloneqq f(x+y)-v(x+y)$ and the following sets:

  * •

$L^{v}\coloneqq L-v=\\{u-v:u\in L\\}$

  * •

$L^{v,y}\coloneqq\\{x\mapsto l(x+y):l\in L^{v}\\}=\\{x\mapsto
u(x+y)-v(x+y):u\in L\\}$

  * •

$L^{v,y}_{0}\coloneqq\\{l-l(0):l\in L^{v,y}\\}=\\{x\mapsto
u(x+y)-v(x+y)+v(y)-u(y):u\in L\\}$.

Then the function $\bar{f}$ is $L^{v,y}$-convex, and more generally
$H_{L^{v,y}_{0}}$-convex. By Remark 2,
$\partial_{L^{v,y}_{0}}\bar{f}(x)=\\{x\mapsto
u(x+y)-u(y)-v(x+y)+v(y):u\in\partial_{L}f(x+y)\\}$. It follows from Equation
(9) that for any $l\in\partial_{L^{v,y}_{0}}(x)$, $l(x)\geq 0$.

Hence, Proposition 16 implies that $0\in\partial_{L^{v,y}_{0}}\bar{f}(0)$.
That is, $\exists u\in\partial_{L}f(0+y)$ such that $u-v-u(y)+v(y)=0$, that
is, $u=v+u(y)-v(y)$.

Therefore, we have that for any $x\in X$,

$f(x)\geq f(y)+(v(x)+u(y)-v(y))-(v(y)+u(y)-v(y))=f(y)+v(x)-v(y).$

Therefore, $v\in\partial_{L}f(y)$ which concludes our proof. ∎

######  Remark 4.

[20, proposition 1.1.11] proved that every maximal cyclic monotone operator is
the subdifferential of some function $f$.

##  5 Conclusions and future research directions

In this paper we obtain summation and composition rules for subdifferential
calculus, and prove that under some reasonable conditions the subdifferential
is a maximal abstract monotone operator. We also highlighted one of the
challenges in the extension of abstract convexity and developing numerical
methods: the fact that it is not always possible to separate two abstract
convex disjoint sets. This leads us to Open Problem 1.

######  Open Problem 1.

What conditions over the set $L$ are sufficient for a Minkovski-like
separation result?

It is natural to assume that, similar to classical convex analysis, the
optimality conditions are formulated in terms of maximal deviations
(Proposition 9). This is where Open Problem 1 appeared. This problem
potentially leads to one more subdifferential calculus rule (maximum).

Finally, this paper identifies one more potential application for abstract
convexity: discrete and integer optimisation.

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††thanks<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
# Observation of topological flat bands in the kagome semiconductor Nb3Cl8

Zhenyu Sun⋆ Institute of Physics, Chinese Academy of Sciences, Beijing 100190,
China  School of Physical Sciences, University of Chinese Academy of Sciences,
Beijing 100049, China  Hui Zhou⋆ Institute of Physics, Chinese Academy of
Sciences, Beijing 100190, China  School of Physical Sciences, University of
Chinese Academy of Sciences, Beijing 100049, China  Cuixiang Wang⋆ Institute
of Physics, Chinese Academy of Sciences, Beijing 100190, China  School of
Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049,
China  Shiv Kumar⋆ Hiroshima Synchrotron Radiation Center, Hiroshima
University, 2-313 Kagamiyama, Higashi-Hiroshima 739-0046, Japan  Daiyu Geng
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
School of Physical Sciences, University of Chinese Academy of Sciences,
Beijing 100049, China  Shaosheng Yue  Institute of Physics, Chinese Academy of
Sciences, Beijing 100190, China  School of Physical Sciences, University of
Chinese Academy of Sciences, Beijing 100049, China  Xin Han  Institute of
Physics, Chinese Academy of Sciences, Beijing 100190, China  School of
Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049,
China  Yuya Haraguchi  Department of Applied Physics and Chemical Engineering,
Tokyo University of Agriculture and Technology, Koganei, Tokyo 184-8588, Japan
Kenya Shimada  Hiroshima Synchrotron Radiation Center, Hiroshima University,
2-313 Kagamiyama, Higashi-Hiroshima 739-0046, Japan  Peng Cheng  Institute of
Physics, Chinese Academy of Sciences, Beijing 100190, China  School of
Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049,
China  Lan Chen  Institute of Physics, Chinese Academy of Sciences, Beijing
100190, China  School of Physical Sciences, University of Chinese Academy of
Sciences, Beijing 100049, China  Songshan Lake Materials Laboratory, Dongguan,
Guangdong 523808, China  Youguo Shi  Institute of Physics, Chinese Academy of
Sciences, Beijing 100190, China  School of Physical Sciences, University of
Chinese Academy of Sciences, Beijing 100049, China  Songshan Lake Materials
Laboratory, Dongguan, Guangdong 523808, China  Center of Materials Science and
Optoelectronics Engineering, University of Chinese Academy of Sciences,
Beijing 100049, China  Kehui Wu  Institute of Physics, Chinese Academy of
Sciences, Beijing 100190, China  School of Physical Sciences, University of
Chinese Academy of Sciences, Beijing 100049, China  Songshan Lake Materials
Laboratory, Dongguan, Guangdong 523808, China  Sheng Meng  Institute of
Physics, Chinese Academy of Sciences, Beijing 100190, China  School of
Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049,
China  Baojie Feng  Institute of Physics, Chinese Academy of Sciences, Beijing
100190, China  School of Physical Sciences, University of Chinese Academy of
Sciences, Beijing 100049, China

###### Abstract

The destructive interference of wavefunctions in a kagome lattice can give
rise to topological flat bands (TFBs) with a highly degenerate state of
electrons. Recently, TFBs have been observed in several kagome metals,
including Fe3Sn2, FeSn, CoSn, and YMn6Sn6. Nonetheless, kagome materials that
are both exfoliable and semiconducting are lacking, which seriously hinders
their device applications. Herein, we show that Nb3Cl8, which hosts a
breathing kagome lattice, is gapped out because of the absence of inversion
symmetry, while the TFBs survive because of the protection of the mirror
reflection symmetry. By angle-resolved photoemission spectroscopy measurements
and first-principles calculations, we directly observe the TFB and a moderate
band gap in Nb3Cl8. By mechanical exfoliation, we successfully obtain
monolayers of Nb3Cl8 and confirm that they are stable under ambient
conditions. In addition, our calculations show that monolayers of Nb3Cl8 have
a magnetic ground state, thus providing opportunities to study the interplay
between geometry, topology, and magnetism.

The interplay between geometry, topology, and magnetism at the quantum level
can give rise to rich physical properties, and the exploration of novel
physics in nontrivial lattices is at the forefront of condensed matter physics
LinZ2018 ; KangM2020NM ; KangM2020NC ; LiuZ2020 ; LiM2021 ; WeeksC2010 ;
SunK2011 ; SlotMR2017 ; Kempkes2019 ; YinJ2020 ; LiuH2021 . A prototypical
example is the kagome lattice that is composed of corner-sharing triangles
(Fig. 1a). Such a simple lattice has been intensively studied because of the
emergence of topological band structures and frustration-driven spin-liquid
states GuoHM2009 ; BalentsL2010 ; TangE2011 ; HanT2012 . Based on a simple
$s$-orbital tight-binding (TB) model with nearest-neighbour hopping, the
electronic band structure of the kagome lattice can be described as a Dirac
cone capped with a TFB (Fig. 1b). The eigenfunctions at neighbouring corners
have opposite phases, which results in a phase cancellation for hopping to
neighbouring sites (black arrows in Fig. 1a). Therefore, the electronic state
is geometrically confined within a single hexagon; this real-space electronic
localization leads to a TFB with quenched kinetic energy. The strong electron
correlation effects in the highly degenerate flat band can give rise to
various exotic properties, including high-temperature superconductivity
ImadaM2000 ; JiangYX2021 , fractional quantum Hall effects TangE2011 , and
Wigner crystal states JiangHC2017 ; JaworowskiB2018 .

Recently, TFBs and Dirac cones have been observed in several kagome metals by
angle-resolved photoemission spectroscopy (ARPES), including Fe3Sn2 LinZ2018 ;
YeL2018 , FeSn KangM2020NM , CoSn KangM2020NC ; LiuZ2020 , and YMn6Sn6 LiM2021
. However, most of the previously discovered kagome materials are metals
without a band gap, which strongly limits their applications in optoelectronic
or logic devices. Therefore, it is highly desirable to realize semiconducting
kagome materials with a moderate band gap. On the other hand, an ideal kagome
lattice is only one layer thick, and the non-negligible interlayer coupling in
bulk materials will (partially) break the intrinsic properties of the kagome
lattice. To date, experimental realization of layered and exfoliable kagome
materials with TFBs is still challenging.

Here, we report a combined experimental and theoretical study on a breathing
kagome material Nb3Cl8. Recently, the existence of TFBs in Nb3Cl8 has been
predicted by a high-throughput screening LiuH2021 , but experimental
investigations of its electronic structure are still lacking. Our ARPES and
optical absorption spectroscopy measurements confirmed the existence of TFBs
and a moderate band gap ($\sim$1.12 eV). By mechanical exfoliation, we
successfully obtained monolayer Nb3Cl8, a key step for device applications. In
addition, Nb3Cl8 is expected to have a magnetic ground state, which implies
the existence of exotic properties arising from the interplay between
magnetism and band topology.

Figure 1: Crystal and electronic structure of the kagome and breathing kagome
lattices. (a, c) Schematic drawing of the kagome lattice and breathing kagome
lattice, respectively. Electrons are confined in the blue shaded hexagons
because of destructive interference, as indicated by the black arrows. Red
dashed lines indicate the mirror axis. (b, d) TB band structure of the kagome
lattice and breathing kagome lattice, respectively. The (breathing) kagome
lattice has a (gapped) Dirac cone and a topological flat band. (e) Three-
dimensional crystal structure of Nb3Cl8. Red and blue balls indicate Nb and Cl
atoms, respectively. (f) The crystal structure of monolayer Nb3Cl8. The Nb
atoms form a breathing kagome lattice, as indicated by the black dashed lines.
(g, h) The calculated band structures of monolayer (g) and bulk (h) Nb3Cl8 in
the paramagnetic state. The parity of the mirror operator along $\Gamma$-M is
labelled by “+” and “-” near each band. The four bands that have been observed
by ARPES measurements are indicated by $\alpha$, $\beta$, $\gamma$, and
$\delta$, respectively.

Single crystals of Nb3Cl8 were grown using PbCl2 as a flux. High-purity Nb
(Alfa Aesar 99.99%) and NbCl5 (Alfa Aesar 99.9%) were mixed at a molar ratio
of 7:8 and placed in an alumina crucible. The crucible was sealed in a quartz
tube under vacuum, heated at 750 ∘C for 150 hours, and cooled to room
temperature naturally. The excess PbCl2 flux was removed by sonicating in hot
water. ARPES measurements were performed at Beamline BL-1 of the Hiroshima
Synchrotron Radiation Center IwasawaH2017 . Clean surfaces required for the
ARPES measurements were obtained by cleaving the samples in situ in an
ultrahigh vacuum chamber. Both the cleavage and the measurements were
performed at room temperature to avoid charging effects. Nb3Cl8 thin flakes
can be exfoliated on various substrates, including SiO2/Si, Au/SiO2/Si, and
sapphire, using blue Nitto tapes. Freshly cleaved surfaces were attached to
the precleaned substrates, followed by carefully peeling off the tape. Thin
flakes that contain few- to monolayer Nb3Cl8 remained on the substrates. The
thicknesses and morphologies of Nb3Cl8 flakes were examined by AFM (Oxford,
Asylum Research Cypher S) in tapping mode. Raman spectra were collected using
a confocal Raman system (Horiba LanRam HR Evolution) with 532-nm laser
excitation. Optical absorption measurements were carried out using a
commercial spectrometer (MStarter ABS, Metatest Corporation).

First-principles calculations based on density functional theory were
performed with the Vienna ab initio simulation package KresseG1993 ;
KresseG1996prb . The projector-augmented wave pseudopotential BlochlPE1994
and Perdew-Burke-Ernzerhof exchange-correlation functional PerdewJP1996  were
used. The energy cutoff of the plane-wave basis was set at 350 eV. For
monolayer Nb3Cl8, the vacuum space was set to be larger than 15 Å. The first
BZ was sampled according to the $\Gamma$-centred scheme. For monolayer (bulk)
Nb3Cl8, we used a $k$ mesh of 6$\times$6$\times$1 (6$\times$6$\times$6) for
structural optimization and 12$\times$12$\times$1 (9$\times$9$\times$9) for
the self-consistent calculations. The positions of the atoms were optimized
until the convergence of the force on each atom was less than 0.01 eV/Å. The
convergence condition of the electronic self-consistent loop was 10-5 eV.

First, we briefly discuss the band structure of the breathing kagome lattice
(Fig. 1c) based on a TB model. The degeneracy of the Dirac point in the
conventional kagome lattice is protected by inversion symmetry. In the
breathing kagome lattice, however, the absence of inversion symmetry will gap
out the Dirac cone, leading to a semiconducting ground state. On the other
hand, destructive phase interference, the key reason for the emergence of
TFBs, is protected by three equivalent mirror axes (red dashed lines in Fig.
1a,c). These mirror symmetries survive in the breathing kagome lattice, and
thus, the breathing kagome lattice also hosts TFBs. A detailed TB analysis is
presented in the Supplementary Information.

Figure 2: Characterization of the basic physical properties of Nb3Cl8. (a)
X-ray diffraction pattern of the Nb3Cl8 single crystal. (b) Temperature
dependence of the magnetic susceptibility ($\chi$) of Nb3Cl8. The red dashed
line indicates the fitting results using the Curie-Weiss function. (c)
Temperature dependence of the heat capacity of Nb3Cl8.

Next, we study a van der Waals material Nb3Cl8, which has a breathing kagome
lattice in each layer MillerGJ1995 ; HaraguchiY2017 . Bulk Nb3Cl8 crystallizes
in a layered structure with a space group of P3̄m1 (Fig. 1e). Within each
layer, the Nb atoms form a breathing kagome lattice, as shown in Fig. 1f. The
calculated band structure of monolayer Nb3Cl8 is displayed in Fig. 1g. One can
directly see the TFBs above and below the Fermi level, as highlighted by the
orange region. The Dirac cones are gapped out, resulting in a semiconducting
ground state. Because of the weak interlayer van der Waals interaction, the
band structure of bulk Nb3Cl8 is analogous to that of monolayer Nb3Cl8, as
shown in Fig. 1h. As a result, the TFBs survive in bulk Nb3Cl8, which is
favourable for ARPES measurements.

High-quality Nb3Cl8 single crystals were synthesized by the self-flux method
(see the Methods section). Figure 2a shows an XRD spectrum measured along the
(_00l_) direction, and only the _00l_ peaks were observed. The sharp peaks
indicate the high quality of the crystals. The temperature-dependent magnetic
susceptibility of Nb3Cl8 is displayed in Fig. 2b. As the temperature
decreases, the magnetic susceptibility of Nb3Cl8 drops abruptly at $\sim$100 K
with a prominent hysteresis, indicating a magnetic to nonmagnetic phase
transition. Above 100 K, the temperature dependence of the magnetic
susceptibility can be well fitted by the Curie-Weiss function, as indicated by
the red dashed line in Fig. 2b, and the fitted $\theta$ value is -18.88 K. The
negative $\theta$ value indicates an antiferromagnetic ground state. These
results agree well with previous works HaraguchiY2017 . In addition, the heat
capacity shows a sharp $\lambda$-shaped peak at approximately $\sim$98 K, in
line with the transition temperature determined from the magnetic
susceptibility measurements. This phase transition were interpreted as a
slight change in layer stacking, which does not affect the structure within
each layer HaraguchiY2017 . Therefore, the topological properties of Nb3Cl8
are not affected by the phase transition either.

We then performed ARPES measurements to confirm the TFBs in Nb3Cl8. Because of
the semiconducting nature of Nb3Cl8, there is no detectable photoemission
signal at the Fermi level. With increasing binding energies, we observe strong
spectral weight near the $\Gamma$ point of the first Brillouin zone (BZ) from
$E\rm_{B}\sim$0.8 eV, as shown in Fig. 3a. The spectral weight in the second
BZ is much weaker because of the photoemission matrix element effect. The
constant energy contour at $E\rm_{B}\sim$2.6 eV shows clear hexagonal
symmetry, in agreement with the crystal structure of Nb3Cl8. The band
structures along the $\Gamma$-K and $\Gamma$-M directions are displayed in
Fig. 3c-i. A careful comparison with the calculation results shows that the
chemical potential has a $\sim$0.8 eV shift towards higher binding energies.
After readjusting the chemical potential, the calculated band structures agree
well with our ARPES results except for a slight discrepancy at the Fermi
level, as indicated by the red dashed lines. The discrepancy at the Fermi
level will be discussed later. Within 3.5 eV of the Fermi level, we observed
four prominent bands: $\alpha$, $\beta$, $\gamma$, and $\delta$. Notably, the
TFB, i.e., the $\gamma$ band can be observed in the whole BZ and has
negligible dispersion, as shown in Fig. 3g,h,j. The $\beta$ band, which is
very close to the $\gamma$ band, has a stronger spectral weight with low
photon energies (Fig. 3i). In contrast with the flat band, the $\beta$ band
disperses with momentum, as shown in Fig. 3k.

Figure 3: ARPES measurements of Nb3Cl8. (a,b) Constant energy contours at
$E\rm_{B}$=0.8 and 2.6 eV, respectively. Green lines indicate the BZs of
monolayer Nb3Cl8. (c-i) ARPES intensity plots along the $\Gamma$-K and
$\Gamma$-M directions with different photon energies and polarizations. The
calculated band structures of monolayer Nb3Cl8 are superimposed on each panel
after shifting the Fermi level 0.8 eV towards a higher binding energy. (j,k)
Energy distribution curves of (f) and (i), respectively. Black dashed lines
indicate the $\gamma$ and $\beta$ bands, respectively.

During ARPES measurements, we find that the spectral weight of each band
varies with the polarization of the incident light: the $\alpha$, $\beta$, and
$\delta$ bands are more sensitive to $p$-polarized light, while the $\gamma$
band is more sensitive to $s$-polarized light, despite the intensity variation
with photon energy. To understand this phenomenon, we focus on the bands along
$\Gamma$-M because $\Gamma$-M is a mirror axis and each band along this
direction has definite odd or even parity when spin-orbit coupling is
neglected. The calculation results are indicated by “+” (even) and “-” (odd)
in Fig. 1g. We find that the $\alpha$, $\beta$, and $\delta$ bands have even
parity, which can only be observed by $p$-polarized light. The $\gamma$ band,
i.e. , the flat band has odd parity and can only be detected by $s$-polarized
light. These analyses agree well with our ARPES results, which further
confirms the existence of TFBs in Nb3Cl8. Neglecting the variation in relative
intensity, we find that all bands have negligible dispersion with photon
energy (or $k_{z}$), in agreement with the two-dimensional nature of Nb3Cl8.

Next, we discuss the discrepancy near the Fermi level between the experimental
and calculation results: the calculation results show the existence of bands
after adjusting the chemical potential, while we did not observe any bands in
our ARPES measurements. A possible reason for this discrepancy is the
underestimation of the band gap in DFT calculations PerdewJP2017 . Based on
our ARPES results, the valence band maximum is located at $\sim$0.9 eV below
the Fermi level, which indicates that the actual band gap is larger than 0.9
eV. To determine the band gap of Nb3Cl8, we performed optical absorption
spectroscopy measurements. Figure 4a shows a typical absorption spectrum of
Nb3Cl8, and several absorption peaks are observed from the ultraviolet to
near-infrared regions. There is one prominent peak at 1.12 eV, which can be
assigned to the transition from valence to conduction bands, as indicated by
the blue arrow in Fig. 4b. Therefore, the optical band gap is $\sim$1.12 eV,
which is approximately 0.23 eV larger than the calculation results. It should
be noted that there is a broad hump centred at $\sim$1600 nm (0.775 eV). This
hump is within the band gap and might originate from defects or edge states
RoxloCB1986 ; ChenJ2021 . Since the valence band maximum is located at
$E\rm_{B}\sim$0.9 eV, the conduction band bottom is expected to be located at
$>$0.22 eV above the Fermi level, which is not accessible in our ARPES
measurements.

Thus far, we have demonstrated that Nb3Cl8 hosts TFBs because of the breathing
kagome lattice. Strictly speaking, however, a kagome lattice refers to
monolayer materials; the non-negligible interlayer coupling in bulk materials
could be detrimental to the intrinsic properties of the kagome lattice.
Therefore, it is highly desirable to obtain monolayer kagome materials for
their applications in quantum devices. Because of the weak van der Waals
interaction in bulk Nb3Cl8, we can easily obtain few- to monolayer samples by
mechanical exfoliation. Figure 4c shows a typical optical microscope image of
an ultrathin Nb3Cl8 flake, and Figure 4d shows an atomic force microscopy
(AFM) image in the white box of Fig. 4c. The height of the thinnest flake is
approximately 0.8 nm with respect to the substrate (Fig. 4e), which is
comparable to the lattice constant of bulk Nb3Cl8 in the perpendicular
direction ($\sim$0.61 nm). The quality of ultrathin Nb3Cl8 was also confirmed
by Raman spectroscopy measurements, as shown in Fig. 4f. Most of the Raman
peaks of the bulk Nb3Cl8 survive in the monolayer limit with negligible shift.
It should be noted that our AFM and Raman spectroscopy measurements were
performed under ambient conditions, and the strong intensity of Raman peaks
indicates the high stability of monolayer Nb3Cl8 in air. Stability in air is
an essential prerequisite for potential device applications.

Figure 4: Optical absorption measurements and mechanical exfoliation of
Nb3Cl8. (a) Extinction spectrum (1-T/T0) of a 100-nm thick Nb3Cl8 flake. The
supporting substrate is sapphire. (b) Calculated local density of states. The
chemical potential and band gap were adjusted according to our experimental
results. The blue arrow indicates the optical transition from the valence band
maximum to the conduction band minimum. (c) Optical microscope image of an
exfoliated Nb3Cl8 flake. (d) AFM image in the white box in (c). (e) Line
profile along the white dashed line in (d). (f) Raman spectra of 1L, 2L, and
bulk Nb3Cl8.

To summarize, we provide compelling evidence for the existence of TFBs in
Nb3Cl8 which has a breathing kagome lattice. The semiconducting nature of
Nb3Cl8 provides opportunities to fabricate optoelectronic and logic devices.
Few- and monolayer Nb3Cl8 can be easily obtained by mechanical exfoliation and
are quite stable under ambient conditions, which is an essential prerequisite
for their device applications. In addition, the ground state of monolayer
Nb3Cl8 is ferromagnetic and the TFBs will spin split because of the magnetic
exchange interaction (see Supplementary Information). In contrast with
conventional and nonmagnetic kagome lattices, the breaking of symmetries in
breathing kagome lattices, including inversion and time reversal, can give
rise to multiple topological states, such as higher-order topological
insulators/semimetals BolensA2019 ; WakaoH2020 ; EzawaM2018 , Chern insulators
RenY2021 , and chiral charge densities JiangYX2021 .

⋆ These authors contributed equally to this work.

###### Acknowledgements.

This work was supported by the Ministry of Science and Technology of China
(Grant No. 2018YFE0202700), the National Natural Science Foundation of China
(Grants No. 11974391, No. 11825405, No. 1192780039, and No. U2032204), the
Beijing Natural Science Foundation (Grant No. Z180007), the International
Partnership Program of Chinese Academy of Sciences (Grant No.
112111KYSB20200012), and the Strategic Priority Research Program of Chinese
Academy of Sciences (Grants No. XDB33030100). ARPES measurements were
performed under the Proposal No. 21AU002. We thank the N-BARD, Hiroshima
University for supplying liquid He.

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