On support points and support functionals of convex sets

On support points and support functionals of convex sets

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 Σ 1 ( K ) of the norm-one support functionals of K is uncountable in each nonempty open set that intersects the dual unit sphere. In par...

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Veröffentlicht in:Israel journal of mathematics 2009-06, Vol.171 (1), p.15-27
Hauptverfasser: De Bernardi, C., Veselý, L.
Format: Artikel
Sprache:eng
Schlagworte:
Algebra
Analysis
Applications of Mathematics
Group Theory and Generalizations
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Theoretical
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Zusammenfassung: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 Σ 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 Σ 1 ( K ) is always uncountable, which answers a question posed by L. Zajíček.
ISSN:0021-2172
1565-8511
DOI:10.1007/s11856-009-0037-6