Given x, a point of a convex subset C of an Euclidean space, the two following statements are proven to be equivalent: (i) any 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 mappings and Fσ domains, we prove that any 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.
Citation
unpublished: Aix-Marseille Universités, december 2011