A converse of the Gale-Klee-Rockafellar theorem: Continuity of convex functions at the boundary of their domains
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- by Emil Ernst PDF
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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.References
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Additional Information
- Emil Ernst
- Affiliation: UMR 6632, Aix-Marseille University, Marseille, F-13397, France
- Email: Emil.Ernst@univ-amu.fr
- 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
- © Copyright 2013 American Mathematical Society
- 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
- MathSciNet review: 3080188