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


Author: Emil Ernst
Journal: Proc. Amer. Math. Soc. 141 (2013), 3665-3672
MSC (2010): Primary 52A20; Secondary 52A41, 52B99
DOI: https://doi.org/10.1090/S0002-9939-2013-11643-6
Published electronically: July 9, 2013
MathSciNet review: 3080188
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Abstract | References | Similar Articles | Additional Information

Abstract: Given $ x_0$, 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\to \mathbb{R}$ is upper semi-continuous at $ x_0$, and (ii) $ C$ is polyhedral at $ x_0$. In the particular setting of closed convex functions and $ F_\sigma $ domains, we prove that every closed convex function $ f:C\to \mathbb{R}$ is continuous at $ x_0$ if and only if $ C$ is polyhedral at $ x_0$. This provides a converse to the celebrated Gale-Klee-Rockafellar theorem.


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Additional Information

Emil Ernst
Affiliation: UMR 6632, Aix-Marseille University, Marseille, F-13397, France
Email: Emil.Ernst@univ-amu.fr

DOI: https://doi.org/10.1090/S0002-9939-2013-11643-6
Keywords: Continuity of convex functions, closed convex functions, polyhedral points, conical points, Gale-Klee-Rockafellar theorem, linearly accessible points
Received by editor(s): December 7, 2011
Received by editor(s) in revised form: January 4, 2012, and January 6, 2012
Published electronically: July 9, 2013
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2013 American Mathematical Society

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