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Let A be an open convex subset of a real normed linear space X. We prove that if the boundary of A contains a non-LUR point then there exists a Lipschitz quasiconvex function f from A to \mathbb{R} R not admitting any continuous quasiconvex extension to the whole X.
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) Σ 1 ( K ) of all norm-one support functionals of K is uncountable....
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