Abstract view
Diametrically Maximal and Constant Width Sets in Banach Spaces


Published:20060801
Printed: Aug 2006
J. P. Moreno
P. L. Papini
R. R. Phelps
Abstract
We characterize diametrically maximal and constant width
sets in $C(K)$, where $K$ is any compact Hausdorff space. These
results are applied to prove that the sum of two diametrically
maximal sets needs not be diametrically maximal, thus solving a
question raised in a paper by Groemer. A~characterization of
diametrically maximal sets in $\ell_1^3$ is also given, providing
a negative answer to Groemer's problem in finite dimensional
spaces. We characterize constant width sets in $c_0(I)$, for
every $I$, and then we establish the connections between the Jung
constant of a Banach space and the existence of constant width
sets with empty interior. Porosity properties of families of sets
of constant width and rotundity properties of diametrically
maximal sets are also investigated. Finally, we present some
results concerning nonreflexive and Hilbert spaces.