Uniform Rotundity with Respect to Finite-Dimensional Subspaces

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 (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 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.