Abstract

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 Σ1(K)\Sigma_1(K) of all norm-one support functionals of K is uncountable. This was proved for bounded K by L. Vesely and the author ["On support points and support functionals of convex sets", Israel J. Math. 171 (2009) 15–27], and for general K by L.Vesely ["A parametric smooth variational principle and support properties of convex sets and functions", J. Math. Anal. Appl. 350 (2009) 550–561] using a parametric smooth variational principle. We present an alternative geometric proof of the general case in the spirit of the paper of the author and L. Vesely cited above.

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Keywords

  • Convex set
  • support point
  • support functional
  • Bishop-Phelps theorem

Mathematics Subject Classification

46A55; 46B99, 52A07

References

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Suggested citation

C. A. De Bernardi. “On Support Points and Functionals of Unbounded Convex Sets.” Journal of Convex Analysis 20 (2013), No. 3, 871–880. https://doi.org/10.68381/jca20046

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Published by Heldermann Verlag, 2013. Rights now held by Banach Press LLC.