A converse of the Gale-Klee-Rockafellar theorem: Continuity of convex functions at the boundary of their domains

Abstract

Given x 0 x_0 , a point of a convex subset C C of a Euclidean space, the two following statements are proven to be equivalent: (i) every convex function f : C → R f:C\to \mathbb {R} is upper semi-continuous at x 0 x_0 , and (ii) C C is polyhedral at x 0 x_0 . In the particular setting of closed convex functions and F σ F_\sigma domains, we prove that every closed convex function f : C → R f:C\to \mathbb {R} is continuous at x 0 x_0 if and only if C C is polyhedral at x 0 x_0 . This provides a converse to the celebrated Gale-Klee-Rockafellar theorem.