
Journal of Convex Analysis 25 (2018), No. 4, 12231252 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 (201 KB)] for subscribers only. 