%0 Journal Article
%T A converse of the Gale-Klee-Rockafellar theorem: Continuity of convex functions at the boundary of their domains
%+ Institut de Mathématiques de Marseille (I2M)
%A Ernst, Emil
%< avec comité de lecture
%@ 0002-9939
%J Proceedings of the American Mathematical Society
%I American Mathematical Society
%V 141
%N 10
%8 2013-10
%D 2013
%R 10.1090/S0002-9939-2013-11643-6
%K continuity of convex functions
%K closed convex functions
%K polyhedral points
%K conical points
%K Gale-Klee-Rockafellar theorem
%K linearly accessible points
%Z Mathematics [math]/Classical Analysis and ODEs [math.CA]Journal articles
%X Given x, a point of a convex subset C of a Euclidean space, the two following statements are proven to be equivalent: (i) every convex function f : C → R is upper semi-continuous at x, and (ii) C is polyhedral at x. In the particular setting of closed convex functions and F-sigma domains, we prove that every closed convex function f : C → R is continuous at x if and only if C is polyhedral at x. This provides a converse to the celebrated Gale - Klee - Rockafellar theorem.
%G English
%L hal-01271975
%U https://hal.science/hal-01271975
%~ CNRS
%~ UNIV-AMU
%~ EC-MARSEILLE
%~ INSMI
%~ I2M
%~ I2M-2014-