We prove that the set of all support points of a nonempty closed convex bounded set C in a real infinite-dimensional Banach space X is AR(σ-compact) and contractible. Under suitable conditions, similar results are proved also for the set of all support functionals ofC and for the domain, the graph, and the range of the subdifferential map of a proper convex lower semicontinuous function on X.
Higher connectedness properties of support points and functionals of convex sets / C. A. De Bernardi. - In: CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES. - ISSN 0008-414X. - 65:6(2013), pp. 1236-1254.
Higher connectedness properties of support points and functionals of convex sets
C. A. De Bernardi
2013
Abstract
We prove that the set of all support points of a nonempty closed convex bounded set C in a real infinite-dimensional Banach space X is AR(σ-compact) and contractible. Under suitable conditions, similar results are proved also for the set of all support functionals ofC and for the domain, the graph, and the range of the subdifferential map of a proper convex lower semicontinuous function on X.
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