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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$.


References [Enhancements On Off] (What's this?)

  • [1] W. Fenchel, On conjugate convex functions, Canad. J. Math. 1 (1949), 73-77. MR 0028365 (10:435b)
  • [2] W. M. Hirsch and A. J. Hoffman, Extreme varieties, concave functions, and the fixed charge problem, Comm. Pure Appl. Math. 14 (1961), 355-369. MR 0131816 (24:A1664)
  • [3] V. Klee, Extremal structure of convex sets. II, Math. Z. 69 (1958), 90-104. MR 0092113 (19:1065b)
  • [4] -, Some characterizations of convex polyhedra, Acta Math. 102 (1959), 79-107. MR 0105651 (21:4390)
  • [5] -, Review of [2], Math. Reviews 24 (1962), 307.

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Additional Information

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

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