On Support Points and Functionals of Unbounded Convex Sets

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 ["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.