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
MathSciNet review: 0230219
Full-text PDF Free Access
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 is a boundedly polyhedral set and is a convex function on the relative interior of such that is bounded on bounded sets, then can be extended in a unique way to a continuous convex function on .
-  W. Fenchel, On conjugate convex functions, Canadian J. Math. 1 (1949), 73–77. MR 0028365
-  Warren M. Hirsch and Alan J. Hoffman, Extreme varieties, concave functions, and the fixed charge problem., Comm. Pure Appl. Math. 14 (1961), 355–369. MR 0131816, https://doi.org/10.1002/cpa.3160140205
-  V. L. Klee Jr., Extremal structure of convex sets. II, Math. Z. 69 (1958), 90–104. MR 0092113, https://doi.org/10.1007/BF01187394
-  Victor Klee, Some characterizations of convex polyhedra, Acta Math. 102 (1959), 79–107. MR 0105651, https://doi.org/10.1007/BF02559569
-  -, Review of , Math. Reviews 24 (1962), 307.
- W. Fenchel, On conjugate convex functions, Canad. J. Math. 1 (1949), 73-77. MR 0028365 (10:435b)
- 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)
- V. Klee, Extremal structure of convex sets. II, Math. Z. 69 (1958), 90-104. MR 0092113 (19:1065b)
- -, Some characterizations of convex polyhedra, Acta Math. 102 (1959), 79-107. MR 0105651 (21:4390)
- -, Review of , Math. Reviews 24 (1962), 307.