Journal of Convex Analysis 13 (2006), No. 3, 823--837
Copyright Heldermann Verlag 2006
New Families of Convex Sets Related to Diametral Maximality
José Pedro Moreno
Dpto. Matemáticas, Facultad de Ciencias, Universidad Autónoma, Madrid 28049, Spain
Pier Luigi Papini
Dip. di Matematica, Piazza Porta S. Donato 5, 40126 Bologna, Italy
Robert R. Phelps
Dept. of Mathematics, Box 354-350, University of Washington, Seattle, WA 98195, U.S.A.
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.
Keywords: Diametrically maximal, constant radius, constant difference, constant width sets, spherical intersection property.
MSC: 52A05, 46B20
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