On support points and support functionals of convex sets (Q836100)

Let \(K\) be a bounded convex set in a Banach space \(X\). If \(x\in K\) and \(x^*\in X^*\backslash\{0\}\) are such that \(x^*(x)=\sup x^*(K)\), then \(x\) is said to be a support point and \(x^*\) a support functional of \(K\). The authors prove that the set of all support points is pathwise connected and the set \(\Sigma_1(K)\) of the norm-one support functionals is uncountable in each nonempty open set that intersects the dual unit sphere. Before, these result were known under different additional conditions. The second result answers a question of L.\,Zajíček. Reviewer's remark. The well-known problem whether \(\Sigma_1(K)\) is always norm connected is still open, even for balls. \textit{L.\,Veselý} [J.~Math.\ Anal.\ Appl.\ 350, 550--561 (2009; Zbl 1162.49006)] proved that, if the Banach space admits a Fréchet smooth renorming, then \(\Sigma_1(K)\) is pathwise connected and locally pathwise connected.