
Journal of Convex Analysis 13 (2006), No. 3, 823837 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 josepedro.moreno@uam.es Pier Luigi Papini Dip. di Matematica, Piazza Porta S. Donato 5, 40126 Bologna, Italy papini@dm.unibo.it Robert R. Phelps Dept. of Mathematics, Box 354350, University of Washington, Seattle, WA 98195, U.S.A. phelps@math.washington.edu 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 [ Fulltextpdf (407 KB)] for subscribers only. 