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Proceedings of the American Mathematical Society

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Automatic continuity of concave functions

Author: Roger Howe
Journal: Proc. Amer. Math. Soc. 103 (1988), 1196-1200
MSC: Primary 90C20; Secondary 26B25, 52A20
MathSciNet review: 955008
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Abstract: A necessary and sufficient condition is given that a semicontinuous, nonnegative, concave function on a finite dimensional closed convex set $ X$ necessarily be continuous at a point $ {x_0} \in X$. Application of this criterion at all points of $ X$ yields a characterization, due to Gale, Klee and Rockafellar, of convex polyhedra in terms of continuity of their convex functions.

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

  • [GKR] D. Gale, V. Klee and R. T. Rockafellar, Convex functions on convex polytopes, Proc. Amer. Math. Soc. 19 (1968), 867-873. MR 0230219 (37:5782)
  • [K] V. Klee, Some characterizations of compact polyhedra, Acta Math. 102 (1959), 79-107. MR 0105651 (21:4390)
  • [R] R. T. Rockafellar, Convex analysis, Princeton Univ. Press, Princeton, N.J., 1970. MR 0274683 (43:445)

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Keywords: Concave function, semicontinuity, continuity
Article copyright: © Copyright 1988 American Mathematical Society

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