CLOSED MEANS CONTINUOUS IFF POLYHEDRAL: A CONVERSE OF THE GKR THEOREM

Emil Ernst (1)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités

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.

Mots clés: continuity of convex functions; closed convex functions; polyhedral points; conical points; Gale-Klee-Rockafellar theorem; linearly accessible points

Domaines: Optimisation et contrôle [math.OC]

Emil Ernst. CLOSED MEANS CONTINUOUS IFF POLYHEDRAL: A CONVERSE OF THE GKR THEOREM. 2011. ⟨hal-00652630⟩