Geodesic
New Families of Convex Sets Related to Diametral Maximality
Journal of convex analysis, Tome 13 (2006) no. 3, pp. 823-837
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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.
Classification : 52A05, 46B20
Mots-clés : Diametrically maximal, constant radius, constant difference, constant width sets, spherical intersection property 
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     title = {New {Families} of {Convex} {Sets} {Related} to {Diametral} {Maximality}},
     journal = {Journal of convex analysis},
     pages = {823--837},
     year = {2006},
     volume = {13},
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J. P. Moreno; P. L. Papini; R. R. Phelps. New Families of Convex Sets Related to Diametral Maximality. Journal of convex analysis, Tome 13 (2006) no. 3, pp. 823-837. http://geodesic.mathdoc.fr/item/JCA_2006_13_3_JCA_2006_13_3_a21/