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 \Sigma_1(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 \Sigma_1(K) is always uncountable, which answers a question posed by L.Zajicek.
On support points and support functionals of convex sets / C.A. De Bernardi, L. Vesely. - In: ISRAEL JOURNAL OF MATHEMATICS. - ISSN 0021-2172. - 171:1(2009 Jun), pp. 15-27. [10.1007/s11856-009-0037-6]
On support points and support functionals of convex sets
C.A. De BernardiPrimo
;L. VeselyUltimo
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 \Sigma_1(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 \Sigma_1(K) is always uncountable, which answers a question posed by L.Zajicek.
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