Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

A converse of the Gale-Klee-Rockafellar theorem: Continuity of convex functions at the boundary of their domains
HTML articles powered by AMS MathViewer

by Emil Ernst PDF
Proc. Amer. Math. Soc. 141 (2013), 3665-3672 Request permission

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
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 52A20, 52A41, 52B99
  • Retrieve articles in all journals with MSC (2010): 52A20, 52A41, 52B99
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