A selection theorem concerning support points of convex sets in a Banach space is proved. As a corollary we obtain the following result. Denote by BCC (X) the metric space of all nonempty bounded closed convex sets in a Banach space X. Then there exists a continuous mapping S from BCC (X) to X such that S(K) is a support point of K for each K in BCC (X). Moreover, it is possible to prescribe the values of S on a closed discrete subset of BCC(X).
De Bernardi, C. A., On support points and continuous extensions, <<ARCHIV DER MATHEMATIK>>, 2009; 93 (4): 369-378. [doi:10.1007/s00013-009-0044-1] [http://hdl.handle.net/10807/113772]
On support points and continuous extensions
De Bernardi, Carlo Alberto
2009
Abstract
A selection theorem concerning support points of convex sets in a Banach space is proved. As a corollary we obtain the following result. Denote by BCC (X) the metric space of all nonempty bounded closed convex sets in a Banach space X. Then there exists a continuous mapping S from BCC (X) to X such that S(K) is a support point of K for each K in BCC (X). Moreover, it is possible to prescribe the values of S on a closed discrete subset of BCC(X).
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