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

Ernst, Emil

doi:10.1090/proc/13558

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

Proc. Amer. Math. Soc. 145 (2017), 4473-4483

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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