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# Higher Connectedness Properties of Support Points and Functionals of Convex Sets

Published:2012-12-29
Printed: Dec 2013
• Carlo Alberto De Bernardi,
Dipartimento di Matematica, Università degli Studi di Milano, Via C. Saldini 50, 20133 Milano, Italy
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## 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 $\mathrm{AR(}\sigma$-$\mathrm{compact)}$ and contractible. Under suitable conditions, similar results are proved also for the set of all support functionals of $C$ and for the domain, the graph and the range of the subdifferential map of a proper convex l.s.c. function on $X$.
 Keywords: convex set, support point, support functional, absolute retract, Leray-Schauder continuation principle
 MSC Classifications: 46A55 - Convex sets in topological linear spaces; Choquet theory [See also 52A07] 46B99 - None of the above, but in this section 52A07 - Convex sets in topological vector spaces [See also 46A55]

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