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Journal of Convex Analysis 25 (2018), No. 4, [final page numbers not yet available]
Copyright Heldermann Verlag 2018



Uniform Rotundity with Respect to Finite-Dimensional Subspaces

M. Veena Sangeetha
Dept. of Mathematics, Indian Institute of Technology, Madras - Chennai 600036, India
veena176@gmail.com

P. Veeramani
Dept. of Mathematics, Indian Institute of Technology, Madras - Chennai 600036, India
pvmani@iitm.ac.in



We introduce the notion of uniform rotundity of a normed space with respect to a finite dimensional subspace as a generalization of uniform rotundity in a direction. We discuss several characterizations of this property and obtain a series of new classes of normed spaces which in a natural way generalize normed spaces that are uniformly rotund in every direction. Indeed for each positive integer k we get normed spaces that are uniformly rotund with respect to every k-dimensional subspace (UREk) with k=1 reducing to uniform rotundity in every direction. Also UREk implies UREk+1 but not conversely. We show that UREk spaces turn out to be exactly those in which the Chebyshev center of a nonempty bounded set is either empty or is of dimension at most k-1 thus extending a well known result of Garkavi. These spaces have normal structure which is a sufficient condition for fixed property for nonexpansive maps on weakly compact convex sets. In addition, there is a common fixed point in the self-Chebyshev center of a weakly compact convex set for the collection of all isometric selfmaps on the set. Uniform rotundity with respect to a finite dimensional subspace is defined based on Sullivan's notion of k-uniform rotundity in the same fashion as uniform rotundity in a direction is based on Clarkson's uniform rotundity. But a characterization of the same in terms of Milman's modulus of k-uniform rotundity is also discussed.

Keywords: Uniform rotundity with respect to finite-dimensional subspaces, k-uniform rotundity, multi-dimensional volumes, Chebyshev centers, asymptotic centers.

MSC: 46B20, 47H09, 47H10

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