
Journal of Convex Analysis 25 (2018), No. 4, [final page numbers not yet available] Copyright Heldermann Verlag 2018 Uniform Rotundity with Respect to FiniteDimensional 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 kdimensional subspace (URE_{k}) with k=1 reducing to uniform rotundity in every direction. Also URE_{k} implies URE_{k+1} but not conversely. We show that URE_{k} 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 k1 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 selfChebyshev 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 kuniform 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 kuniform rotundity is also discussed. Keywords: Uniform rotundity with respect to finitedimensional subspaces, kuniform rotundity, multidimensional volumes, Chebyshev centers, asymptotic centers. MSC: 46B20, 47H09, 47H10 [ Fulltextpdf (387 KB)] for subscribers only. 