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Closed means continuous iff polyhedral: a converse of the GKR theorem

Emil ERNST(emil.ernst***at***univ-cezanne.fr)

Abstract: 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.

Keywords: continuity of convex functions, closed convex functions, polyhedral points, conical points, Gale-Klee-Rockafellar theorem, linearly accessible points

Category 1: Convex and Nonsmooth Optimization (Convex Optimization )

Citation: unpublished: Aix-Marseille UniversitÚs, december 2011

Download: [PDF]

Entry Submitted: 12/15/2011
Entry Accepted: 12/16/2011
Entry Last Modified: 12/15/2011

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