Abstract

Eggleston proved in a landmark monograph that, in every finite dimensional normed space, a bounded closed convex set with constant radius from its boundary is diametrically maximal. We show that this is no longer true in general and we characterize a set with constant radius by means of an equation involving its radius and diameter. A somewhat similar equation yields the definition of a constant difference set, a notion which turns out to be stronger than diametrically maximal but weaker than constant width. We investigate the interplay of these notions with the geometry of the underlying Banach space.

Author information

Contact details are reproduced from the original publication and may be historical.

José Pedro Moreno

Dpto. Matemáticas, Facultad de Ciencias, Universidad Autónoma, Madrid 28049, Spain

josepedro.moreno@uam.es

Pier Luigi Papini

Dip. di Matematica, Piazza Porta S. Donato 5, 40126 Bologna, Italy

papini@dm.unibo.it

Keywords

  • Diametrically maximal
  • constant radius
  • constant difference
  • constant width sets
  • spherical intersection property

Mathematics Subject Classification

52A05, 46B20

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Suggested citation

J. P. Moreno, P. L. Papini, R. R. Phelps. “New Families of Convex Sets Related to Diametral Maximality.” Journal of Convex Analysis 13 (2006), No. 3/4, 823–837. https://doi.org/10.68381/jca13061

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Published by Heldermann Verlag, 2006. Rights now held by Banach Press LLC.