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 [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].