Let K be a nonempty closed convex subset of a real Banach space of dimension at least two. Suppose that K does not contain any hyperplane. Then the set of all support points of K is pathwise connected and the set Sigma(1)(K) of all norm-one support functionals of K is uncountable. This was proved for bounded K by L. Vesely and the author [3], and for general K by L. Vesely [8] using a parametric smooth variational principle. We present an alternative geometric proof of the general case in the spirit of [3].
De Bernardi, C. A., On Support Points and Functionals of Unbounded Convex Sets, <<JOURNAL OF CONVEX ANALYSIS>>, 2013; 20 (3): 871-880 [http://hdl.handle.net/10807/113768]
On Support Points and Functionals of Unbounded Convex Sets
De Bernardi, Carlo Alberto
2013
Abstract
Let K be a nonempty closed convex subset of a real Banach space of dimension at least two. Suppose that K does not contain any hyperplane. Then the set of all support points of K is pathwise connected and the set Sigma(1)(K) of all norm-one support functionals of K is uncountable. This was proved for bounded K by L. Vesely and the author [3], and for general K by L. Vesely [8] using a parametric smooth variational principle. We present an alternative geometric proof of the general case in the spirit of [3].
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