Continuity of convex functions at the boundary of their domains: an infinite dimensional Gale-Klee-Rockafellar theorem

Ernst, Emil

doi:10.1090/proc/13558

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

by Emil Ernst PDF

Proc. Amer. Math. Soc. 145 (2017), 4473-4483 Request permission

Abstract:

Given $C$ a closed convex set spanning the real Banach space $X$ and $x_0$ a boundary point of $C$, this article proves that the two following statements are equivalent: (i) any lower semi-continuous convex function $f:C\to \mathbb {R}$ is continuous at $x_0$, and (ii) at $x_0$, $C$ is Maserick polyhedral; that is, $C$ is locally the intersection of a finite family of closed half-spaces.

References

Mordecai Avriel, Walter E. Diewert, Siegfried Schaible, and Israel Zang, Generalized concavity, Classics in Applied Mathematics, vol. 63, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2010. Reprint of the 1988 original [ MR0927084]. MR 3396214, DOI 10.1137/1.9780898719437.ch1
J. Bourgain, Strongly exposed points in weakly compact convex sets in Banach spaces, Proc. Amer. Math. Soc. 58 (1976), 197–200. MR 415272, DOI 10.1090/S0002-9939-1976-0415272-3
Michele Conforti, Gérard Cornuéjols, and Giacomo Zambelli, Integer programming, Graduate Texts in Mathematics, vol. 271, Springer, Cham, 2014. MR 3237726, DOI 10.1007/978-3-319-11008-0
Roland Durier and Pier Luigi Papini, Polyhedral norms and related properties in infinite-dimensional Banach spaces: a survey, Atti Sem. Mat. Fis. Univ. Modena 40 (1992), no. 2, 623–645. MR 1200312
Emil Ernst, A converse of the Gale-Klee-Rockafellar theorem: continuity of convex functions at the boundary of their domains, Proc. Amer. Math. Soc. 141 (2013), no. 10, 3665–3672. MR 3080188, DOI 10.1090/S0002-9939-2013-11643-6
Vladimir P. Fonf and Libor Veselý, Infinite-dimensional polyhedrality, Canad. J. Math. 56 (2004), no. 3, 472–494. MR 2057283, DOI 10.4153/CJM-2004-022-7
David Gale, Victor Klee, and R. T. Rockafellar, Convex functions on convex polytopes, Proc. Amer. Math. Soc. 19 (1968), 867–873. MR 230219, DOI 10.1090/S0002-9939-1968-0230219-6
Roger Howe, Automatic continuity of concave functions, Proc. Amer. Math. Soc. 103 (1988), no. 4, 1196–1200. MR 955008, DOI 10.1090/S0002-9939-1988-0955008-9
Robert C. James, Weakly compact sets, Trans. Amer. Math. Soc. 113 (1964), 129–140. MR 165344, DOI 10.1090/S0002-9947-1964-0165344-2
Victor Klee, Some characterizations of convex polyhedra, Acta Math. 102 (1959), 79–107. MR 105651, DOI 10.1007/BF02559569
Joram Lindenstrauss, On operators which attain their norm, Israel J. Math. 1 (1963), 139–148. MR 160094, DOI 10.1007/BF02759700
P. H. Maserick, Convex polytopes in linear spaces, Illinois J. Math. 9 (1965), 623–635. MR 182914
R. R. Phelps, Dentability and extreme points in Banach spaces, J. Functional Analysis 17 (1974), 78–90. MR 0352941, DOI 10.1016/0022-1236(74)90005-6
S. L. Troyanski, On locally uniformly convex and differentiable norms in certain non-separable Banach spaces, Studia Math. 37 (1970/71), 173–180. MR 306873, DOI 10.4064/sm-37-2-173-180