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)(X) the metric space of all nonempty bounded closed convex sets in a Banach space X. Then there exists a continuous mapping S : BCC(X) ® XS:(X)X such that S(K) is a support point of K for each K Î BCC(X)K(X). Moreover, it is possible to prescribe the values of S on a closed discrete subset of BCC(X)(X).
On support points and continuous extensions / C.A. De Bernardi. - In: ARCHIV DER MATHEMATIK. - ISSN 0003-889X. - 93:4(2009), pp. 369-378. [10.1007/s00013-009-0044-1]
On support points and continuous extensions
C.A. De BernardiPrimo
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)(X) the metric space of all nonempty bounded closed convex sets in a Banach space X. Then there exists a continuous mapping S : BCC(X) ® XS:(X)X such that S(K) is a support point of K for each K Î BCC(X)K(X). Moreover, it is possible to prescribe the values of S on a closed discrete subset of BCC(X)(X).
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