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

Emil Ernst (1)

1 I2M - Institut de Mathématiques de Marseille

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.

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

Domaines: Analyse classique [math.CA]

Emil Ernst. A converse of the Gale-Klee-Rockafellar theorem: Continuity of convex functions at the boundary of their domains. Proceedings of the American Mathematical Society, 2013, 141 (10), ⟨10.1090/S0002-9939-2013-11643-6⟩. ⟨hal-01271975⟩