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Continuity of convex functions at the boundary of their domains: an infinite dimensional Gale-Klee-Rockafellar theorem


Author: Emil Ernst
Journal: Proc. Amer. Math. Soc. 145 (2017), 4473-4483
MSC (2010): Primary 52A07; Secondary 52B99, 49N15
DOI: https://doi.org/10.1090/proc/13558
Published electronically: May 4, 2017
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Abstract: Given $ C$ a closed convex set spanning the real Banach space $ X$ and $ x_0$ a boundary point of $ C$, this article proves that the two following statements are equivalent: (i) any lower semi-continuous convex function $ f:C\to \mathbb{R}$ is continuous at $ x_0$, and (ii) at $ x_0$, $ C$ is Maserick polyhedral; that is, $ C$ is locally the intersection of a finite family of closed half-spaces.


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Additional Information

Emil Ernst
Affiliation: Aix-Marseille Université, UMR6632, Marseille, F-13397, France
Email: Emil.Ernst@univ-amu.fr

DOI: https://doi.org/10.1090/proc/13558
Keywords: Continuity of convex functions, polyhedral cones, infinite dimensional polyhedrality, Gale-Klee-Rockafellar theorem
Received by editor(s): November 26, 2015
Received by editor(s) in revised form: October 23, 2016
Published electronically: May 4, 2017
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2017 American Mathematical Society

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