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Counterexamples Concerning Support Theorems for Convex Sets in Hilbert Space

Published online by Cambridge University Press:  20 November 2018

R. R. Phelps
R. R. Phelps*
Affiliation:
Department of Mathematics GN-50, University of Washington, Seattle WA 98195
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Abstract

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The Bishop-Phelps theorem guarantees the existence of support points and support functionals for a nonempty closed convex subset of a Banach space; equivalently, it guarantees the existence of subdifferentials and points of subdifferentiability of a proper lower semicontinuous convex function on a Banach space. In this note we show that most of these results cannot be extended to pairs of convex sets or functions, even in Hilbert space. For instance, two proper lower semicontinuous convex functions need not have a common point of subdifferentiability nor need they have a subdifferential in common. Negative answers are also obtained to certain questions concerning density of support points for the closed sum of two convex subsets of Hilbert space.

Keywords

46B2046B2246C0552A05

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 31 , Issue 1 , 01 March 1988, pp. 121 - 128
Copyright
Copyright © Canadian Mathematical Society 1988

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