Journal of Convex Analysis 20 (2013), No. 3, 871--880
Copyright Heldermann Verlag 2013
On Support Points and Functionals of Unbounded Convex Sets
Carlo Alberto De Bernardi
Dipartimento di Matematica, Università di Milano, Via C. Saldini 50, 20133 Milano, Italy
carloalberto.debernardi@gmail.com
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 Σ1(K) of all norm-one support functionals of K is uncountable. This was proved for bounded K by L. Vesely and the author ["On support points and support functionals of convex sets", Israel J. Math. 171 (2009) 15--27], and for general K by L.Vesely ["A parametric smooth variational principle and support properties of convex sets and functions", J. Math. Anal. Appl. 350 (2009) 550--561] using a parametric smooth variational principle. We present an alternative geometric proof of the general case in the spirit of the paper of the author and L. Vesely cited above.
Keywords: Convex set, support point, support functional, Bishop-Phelps theorem.
MSC: 46A55; 46B99, 52A07
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