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PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 141, Number 10, October 2013, Pages 3665–3672
S 0002-9939(2013)11643-6
Article electronically published on July 9, 2013
ACONVERSEOFTHEGALE-KLEE-ROCKAFELLAR
THEOREM: CONTINUITY OF CONVEX FUNCTIONS
AT THE BOUNDARY OF THEIR D O M A I N S
EMIL ERNST
(Communicated by Thomas Schlumprecht)
Abstract. Given x0,apointofaconvexsubsetCof a Euclidean space,
the two following statements are proven to be equivalent: (i) every convex
function f:C→Ris upper semi-continuous at x0,and(ii)Cis polyhedral
at x0.IntheparticularsettingofclosedconvexfunctionsandFσdomains,
we prove that every closed convex function f:C→Ris continuous at x0if
and only if Cis polyhedral at x0.Thisprovidesaconversetothecelebrated
Gale-Klee-Rockafellar theorem.
1. Introduction
One basic fact about real-valued convex functions on Euclidean spaces is that
they are continuous at each point of their domain’s relative interior (see for instance
[14, Theorem 10.1]).
On the other hand, it is not difficult to define a convex function which is discon-
tinuous at every point of the relative boundary of its domain. As stated by Carter
in his treatise “Foundations of Mathematical Economics” [4, page 334]: This is not
a mere curiosity. Economic life often takes place at the boundaries of convex sets,
where the possibility of discontinuities must be taken into account.
The celebrated Gale-Klee-Rockafellar (GKR) theorem ([6, Theorem 2]; see also
[14, Theorem 10.2]) is a major step toward an accurate understanding of continuity
properties for convex functions at points belonging to the relative boundary of their
domain. This result is particularly meaningful when applied to the class of closed
convex functions, as defined in the seminal work of W. Fenchel ([5]).
GKR Theorem. Aconvexfunctionisuppersemi-continuousateverypointat
which its domain is polyhedral. Accordingly, a closed convex function is continuous
at each such point.
Besides its intrinsic interest, this theorem has proved itself a fertile source of ap-
plications. Taking one example out of many, let us remark that since a polyhedron
is everywhere polyhedral, the GKR theorem proves the ubiquitous mathematical
economics and game theory lemma ([2, Theorem 4.2]), which says that each concave
function defined on Pn
+, the cone of the vectors from Rnwith positive coordinates,
is lower semi-continuous.
Received by the editors December 7, 2011 and, in revised form, January 4, 2012 and January 6,
2012.
2010 Mathematics Subject Classification. Primary 52A20; Secondary 52A41, 52B99.
Key words and phrases. Continuity of convex functions, closed convex functions, polyhedral
points, conical points, Gale-Klee-Rockafellar theorem, linearly accessible points.
c
!2013 American Mathematical Society
3665
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3666 EMIL ERNST
The GKR theorem also provides powerful tools in establishing continuity of
special convex functions issued from particular optimization problems, such as the
M-convex and L-convex functions of Murota and Shioura ([13]).
The example of the closed convex function
f:C→R,f(x, y)= x2
yif (x, y)"=(0,0) and f(0,0) = 0
defined on the disk
C={(x, y)∈R2:x2+(1−y)2≤1},
yet discontinuous at the point (0,0) ∈C, is well-known ([14, page 83]).
Let us remark that the point at which the previously defined function is dis-
continuous may (inter alia) be characterized as being the limit of a non-constant
sequence made of extreme points of the disk. The lemma on page 870 in [6] proves
that this is a very general feature; more precisely, its proof can easily be modified
in order to state the following result.
Converse GKR Theorem. Let Cbe a closed and convex s ubset of X,andx0∈C
be the limit of a non-constant sequence of extreme points of C(such a point exists
if and only if Cis not polyhedral at each and every one of its points). Then there
exists at least one closed convex function f:C→Rwhich is not continuous at x0.
A standard observation proves that if Cis conical at some point x0∈C,then
none of the non-constant sequences of extreme points of Ccan converge to x0.In
this respect, the following result by Howe ([8, Proposition 2]) provides an extension
of the reciprocal GKR theorem.
Howe’s Theorem. Let Cbe a closed and convex subset of Xand x0∈Cbe a
point at which Cis not conical. Then there exists at least one closed convex function
f:C→Rwhich is not continuous at x0.
An obvious limitation of the previous theorem is that Howe’s result is bound to
the setting of closed domains, and no conclusion can be drawn for the larger class
of convex domains over which closed convex functions may be defined (that is, Fσ
convex sets).
Moreover, this result leaves unanswered the decidedly non-trivial question of the
continuity of a closed convex function at points at which the domain is conical
without being polyhedral (typically the apex of a circular cone). Indeed, the hy-
pothesis that a closed convex function is automatically continuous at such types of
points seems very natural, and this claim has been made (in an implicit form) at
least once ([3, Proposition 5, p. 183]). However, this conjecture has been proved
false when Goossens ([7, p. 609]) provided a (very elaborate) example of a closed
convex function defined on a circular cone and discontinuous at its apex.
This note attempts to fill in the gap between the direct GKR theorem and Howe’s
result by proving (Theorem 2.4, Section 2) the following statement.
Second Converse GKR Theorem. Given C,aconvexsubsetoftheEuclidean
space X,andx0,apointatwhichCis not polyhedral, then there is a convex
function f:C→Rwhich is not upper semi-continuous at x0.
When, in addition, Cis a Fσset, then there is f:C→R,aclosedconvex
function which is discontinuous at x0.
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CONVEX FUNCTIONS AT THE BOUNDARY OF THEIR DOMAIN 3667
1.1. Definitions and notation. Let us consider X, a Euclidean space endowed
with the usual topology, and let us set x·yfor the scalar product between the
vectors xand yof X,and&·&for the associated norm.
Given Aa subset of X, let XAbe its affine span (that is, the intersection of all
the hyperplanes of Xcontaining A). The relative boundary of Ais defined by the
formula
r∂(A)=A∩XA\A,
where a superposed bar denotes the closure of a set, while the relation
ri(A)=A\r∂(A)
defines the relative interior of the set A. Let us recall ([14, Theorem 6.2, p. 45])
that the relative interior of a non-empty convex set is non-empty.
As is customary, a subset Aof Xis said to be an Fσset if it is the countable
union of a family of closed subsets of X:
A=
∞
!
i=1
Ai,A
i=Ai∀i∈N.
Obviously, a closed and convex set is Fσ.Inordertoprovideanexampleofaconvex
subset of Xwhich is not Fσ, let us recall ([1, exercise 145, p. 103, and exercise 269,
p. 145]) that the set of the irrationals is not Fσ.Byhomeomorphism,thesame
holds in R2for the set of points on the unit circle possessing an irrational angular
coordinate.
On the other hand, it is an easy exercise to show that the union between an open
disk and an arbitrary subset of its boundary is convex. Therefore, the subset of R2
obtained by putting together the open unit disk and all the points of its boundary
possessing an irrational angular coordinate is a convex set but not an Fσone.
A function f:A→Ris called closed if its epigraph
epi f={(x, r)∈A×R:f(x)≤r}
is a closed subset of X×R.Letusnoticethatthedomainofaclosedfunctionis
necessarily an Fσset. The function f:A→Ris upper semi-continuous at x0if
f(x0)≥lim sup
x∈A, x→x0
f(x).
In this article, by a polyhedron we mean a set obtained as the intersection of
a finite family of closed half-spaces of X; accordingly, polyhedra are closed convex
sets, not always bounded. Following Klee ([10, p. 86]), we call the set Apolyhedral
at x0∈Aif there are U, a neighborhood of x0,andB, a polyhedron, such that
A∩U=A∩B.
Similarly, we call a set Aconical at x0∈Aif there are U, a neighborhood of x0,
and K, a shifted closed convex cone (meaning that 0 is not necessarily its apex),
such that
A∩U=A∩K.
In other words (Howe, [8, p. 1198]), “near x0,thesetAlooks like a [...] cone.”
Obviously, a convex set is polyhedral at each point of its relative interior. Moreover,
if a set is polyhedral at some point, it is also conical at the same point, but the
converse does not generally hold.
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3668 EMIL ERNST
2. Continuity of convex functions at points of the relative
boundary of their domain
A key step in proving our main result is provided by Theorem 2.2. This result
features a geometrical property of points belonging to the relative boundary of a
convex set, which, to the best of our knowledge, has never been addressed.
Following Klee ([9, p. 448]), we call a point x∈Xlinearly accessible from the
subset Aof Xif there is a point asuch that the half-open segment [a;x[ is contained
in A. Of course, the linearly accessible points belong to the closure of A, but the
converse does not generally hold.
For convex sets, however, all points of the closure are linearly accessible (an
obvious application of the fact that their relative interior is always non-empty).
Theorem 2.2 addresses the question of the linear accessibility of the boundary points
for sets which can be expressed as the difference between two convex sets.
Let us first establish to what extent studying this topic helps to demonstrate the
converse GKR theorem.
Proposition 2.1. Let Cbe a subset of Xand x0one of its points, and assume
that there is a closed convex set Dcontaining x0such that x0∈C\D,yetx0is
not linearly accessible from C\D.
i) If Cis convex, then there is a convex function f:C→Rwhich is not upper
semi-continuous at x0.
ii) If Cis a Fσconvex set, then it is possible to find a closed convex function
f:C→Rwhich is not continuous at x0.
Proof of Proposition 2.1.Without loss of generality, we can assume that x0=0.
Let us consider the cone spanned by D,
CD={x∈X:λx∈Dfor some λ>0},
and µD:CD→R,theMinkowskigaugeofD,
µD(x)=inf"γ>0: 1
γx∈D#.
It is clear that CDis a convex cone and µD(0) = 0. Moreover, it is well-known
(see for instance [14, Corollary 9.7.1, p. 79]) that µDis a closed convex function.
We claim that C⊂C
Dand that the restriction
f:C→R,f(x)=µD(x)
of µDto Cfulfills point i) in Proposition 2.1.
Indeed, let x∈C. As 0 is not linearly accessible from C\D, it follows in
particular that the segment [x; 0[ is not entirely contained in C\D,andsince
[x;0[⊂C, it results that
λx∈Dfor some 0 <λ<1;
that is, x∈C
D. Hence, C⊂C
D.
To the end of analyzing the upper semi-continuity of the function fat 0, let us
recall, on one hand, that f(0) = 0 and, on the other, that the point 0 belongs to
the closure of the set C\D.Onecanthusfindasequence,say(xn)n∈N,ofelements
from C\Dconverging to 0. Pick any of the vectors xn. As it does not belong to D,
the definition of the Minkowski gauge implies that f(xn)≥1. The lack of upper
semi-continuity of fat 0 is therefore established.
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CONVEX FUNCTIONS AT THE BOUNDARY OF THEIR DOMAIN 3669
In order to address point ii) of Proposition 2.1, let us state the standard convex
analysis result saying that, given CaconvexFσset, there exists at least one closed
convex function g:C→R. Indeed, Cmay be expressed as the union of an
increasing sequence of convex compact sets, say (An)n∈N,andwemaysetgfor the
convex envelope of the function
h:C→R,h(x) = (min{n∈N:x∈An})2.
Obviously, gis defined on the convex set C.Moreover,thetechniqueusedbyKlee
to prove [10, Theorem 4.1] may be used with virtually no modifications to establish
that gis closed.
If the function gis discontinuous at 0, then it fulfills point ii). Assume now that
the function gis continuous at 0. The application
f:C→R,f(x)=g(x)+µD(x)
is closed and convex as the sum of closed and convex functions. Moreover, fis the
sum between a function which is continuous at 0 and a function which is discontin-
uous at the same point. Thus fis a closed convex application discontinuous at 0,
and Proposition 2.1 is completely proved. !
With the conclusions of Proposition 2.1 in mind, let us address Theorem 2.2, the
most technical part of our paper.
Theorem 2.2. Let Cbe a convex subset of X,andx0be one of its points. The
two following statements are equivalent:
i) Cis not polyhedral at x0.
ii) There is a closed convex set Dcontaining x0such that x0∈C\D,yetx0is
not linearly accessible from C\D.
Proof of Theorem 2.2.Without loss of generality, we can assume that x0=0.
i)⇒ii)LetusassumethatCis not polyhedral at 0. By virtue of Corollary 3.3
in ([10, p. 88]), it results that the same holds for CC, the convex cone spanned by
C.Letusfirstproveageneralresultonnon-polyhedralcones. !
Lemma 2.3. Let Ebe a non-polyhedral convex cone with apex 0.Thenthereisa
sequence (yn)n∈N⊂Xsuch that:
i) for each x∈Eand nlarge enough, the sequence (x·yn)n∈Ntakes only non-
positive values;
ii) for each n∈N,thereisxn∈Esuch that xn·yn>0.
Proof of Lemma 2.3.A far-reaching characterization of polyhedrality for cones was
achieved by Klee ([10, Theorem 4.11, p. 92]; the particular case of closed convex
cones had previously been provided by Mirkil [12, Theorem, p. 1]), which says that
a convex cone is polyhedral if and only if its projection on every two-dimensional
affine manifold of Xis a closed set.
Accordingly, the convex cone Π(E) is not closed, where Π:X→X1is the
operator of projection onto some plane X1of X.Letvbe a vector belonging to
the closure of Π(E)butnottoΠ(E) itself (as 0 = Π(0), it follows that v"=0).
As the relative interiors of a convex set and of its closure coincide, the fact that
the vector vbelongs to Π(E)\Π(E) implies that vlies within the relative boundary
of Π(E). A standard supporting hyperplane argument shows that there exists an
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3670 EMIL ERNST
element w∈X1such that the function x→x·wachieves its maximum over Π(E)
at v. On one hand, this fact implies that
(2.1) Π(x)·w≤v·w∀x∈E;
on the other, since both the vectors 0 and 2 vbelong to Π(E), it yields that
0=0·w≤v·w, 2v·w≤v·w.
Hence
(2.2) v·w=0.
For eve r y n∈N, let us set yn=w+v
n.Asyn∈X1, it results that
(2.3) x·yn=Π(x)·$w+v
n%∀x∈E, n ∈N.
We claim that the sequence (yn)n∈Nfulfills relation i). Let us pick x∈E.In
view of relation (2.1), there are two possible cases: a)Π(x)·w<v·wand b)
Π(x)·w=v·w.
In case a), from relation (2.2) we infer that
(2.4) Π(x)·w<0.
As obviously
(2.5) lim
n→∞ Π(x)·v
n=0,
statement i) comes from relations (2.3), (2.4) and (2.5).
In case b), Π(x)·w=0,soΠ(x) belongs to the hyperplane
Hw={z∈X1:z·w=0}
of X1.ButX1is a two-dimensional vector space, and each of its hyperplanes is
in fact a line; moreover, we have already proved (relation (2.2)) that the non-null
vector vbelongs to Hw.ItresultsthatΠ(x)liesonthelineRv.
As v/∈Π(E)andsinceΠ(E) is a cone with apex 0, it follows that the half-line
R∗
+vis disjoint from Π(E). We may thus affirm that
(2.6) Π(x)=−λvfor some λ≥0.
By combining relations (2.2), (2.3) and (2.6), we conclude that
x·yn=−λ&v&2
n≤0∀n∈N.
Statement i) is therefore fulfilled in both situations a)andb).
Let us now address relation ii). As v∈Π(E), there is a sequence (zn)n∈N⊂E
such that the sequence (Π(zn))n∈Nconverges to v. Pick k∈Nand apply relation
(2.3) for x=znand yk:
zn·yk=Π(zn)·$w+v
k%∀n∈N.
Accordingly,
lim
n→∞ (zn·yk)=v·$w+v
k%;
by virtue of relation (2.2), we obtain that
lim
n→∞ (zn·yk)=&v&2
k>0.
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CONVEX FUNCTIONS AT THE BOUNDARY OF THEIR DOMAIN 3671
The set Lk={n∈N:zn·yk>0}is therefore non-empty. Set u(k) = min Lk,
and, for each positive integer n,definexnas being zu(n). The sequences (xn)n∈N
and (yn)n∈Nobviously fulfill relation ii). !
Let us now get back to the proof of the implication i)⇒ii) from Theorem 2.2
and apply the conclusions of Lemma 2.3 to the non-polyhedral cone CC.
Accordingly, there are two sequences, (xn)n∈N⊂C
Cand (yn)n∈N⊂X,suchthat
(2.7) ∀x∈C
C,x·yn≤0fornlarge enough,
and
(2.8) xn·yn>0∀n∈N.
Since xnis a vector from the cone spanned by the convex set Cand 0 ∈C,wefind
that there is a positive real number, say ζn,suchthat
λx
n∈C∀0≤λ≤ζn.
Set
λn= min &ζn,1
n&xn&'∀n∈N.
It follows that λnxn∈Cfor each positive integer nand that the sequence (λnxn)n∈N
converges to 0.
Define the set
D={x∈X:x·yn≤λn
xn·yn
2∀n∈N}.
Obviously, Dis a closed convex set which contains the point 0. Moreover, for every
positive integer n,thepointλnxnbelongs to Cbut does not belongs to D.Asthe
sequence λnxnconverges to 0, we may conclude that 0 ∈C\D.
To show that 0 is not linearly accessible from C\D, let us pick x∈Cand recall
(see relation (2.7)) that the sequence (x·yn)n∈Ntakes only a finite number of pos-
itive values, while, by virtue of the inequality (2.8), the sequence $λnxn·yn
2%n∈N
has only positive terms. Thus, the sequence &2(x·yn)
λn(xn·yn)'n∈N
has a finite number
of positive values, so there is a positive real number asuch that
2(x·yn)
λn(xn·yn)≤a∀n∈N.
For every positive number µsuch that µa≤1, it results that
µ(x·yn)≤(µa)λn
xn·yn
2≤λn
xn·yn
2∀n∈N.
Accordingly,
µx∈D∀0≤µ≤1
a,
and we may conclude that there is no point x∈Csuch that the segment [x, 0[ is
entirely contained within C\D. In other words, the point 0 is not linearly accessible
from C\D.
ii)⇒i) This implication easily follows by combining the classical GKR theorem
and Proposition 2.1. !
The main result of this note now stems from combining Proposition 2.1 and
Theorem 2.2.
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3672 EMIL ERNST
Theorem 2.4. Given C,aconvexsubsetoftheEuclideanspaceX,andx0∈C,
every convex function f:C→Ris upper semi-continuous at x0if and only if C
is polyhedral at x0.
When, in addition, Cis an Fσset, then each closed convex function f:C→R
is continuous at x0if and only if Cis polyhedral at x0.
Acknowledgment
The author is grateful to the anonymous referee for helpful comments and sug-
gestions, which have been included in the final version of this paper.
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UMR 6632, Aix-Marseille University, Marseille, F-13397, France
E-mail address:Emil.Ernst@univ-amu.fr
