Let K be a bounded closed convex subset of a real Banach space of dimension at least two. Then the set of the support points of K is pathwise connected and the set NA1(K) of the norm-one support functionals of K is uncountable in each nonempty open set that intersects the dual unit sphere. In particular, the set NA1(K) is always uncountable, which answers a question posed by L. Zajicek.
De Bernardi, C. A., Veselý, L., On support points and support functionals of convex sets, <<ISRAEL JOURNAL OF MATHEMATICS>>, 2009; 171 (1): 15-27. [doi:10.1007/s11856-009-0037-6] [http://hdl.handle.net/10807/113777]
On support points and support functionals of convex sets
De Bernardi, Carlo Alberto;
2009
Abstract
Let K be a bounded closed convex subset of a real Banach space of dimension at least two. Then the set of the support points of K is pathwise connected and the set NA1(K) of the norm-one support functionals of K is uncountable in each nonempty open set that intersects the dual unit sphere. In particular, the set NA1(K) is always uncountable, which answers a question posed by L. Zajicek.
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