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Proceedings of the American Mathematical Society

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

 

 

Convex functions on convex polytopes


Authors: David Gale, Victor Klee and R. T. Rockafellar
Journal: Proc. Amer. Math. Soc. 19 (1968), 867-873
MSC: Primary 52.10; Secondary 90.00
DOI: https://doi.org/10.1090/S0002-9939-1968-0230219-6
MathSciNet review: 0230219
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Abstract: The behavior of convex functions is of interest in connection with a wide variety of optimization problems. It is shown here that this behavior is especially simple, in certain respects, when the domain is a polytope or belongs to certain classes of sets closely related to polytopes; moreover, the polytopes and related classes are actually characterized by this simplicity of behavior.

The following corollary is useful in mathematical economics: If $ D$ is a boundedly polyhedral set and $ \phi $ is a convex function on the relative interior of $ D$ such that $ \phi $ is bounded on bounded sets, then $ \phi $ can be extended in a unique way to a continuous convex function on $ D$.


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DOI: https://doi.org/10.1090/S0002-9939-1968-0230219-6
Article copyright: © Copyright 1968 American Mathematical Society