On k-rotundity and k-uniform rotundity in direct sums of normed spaces

Abstract

A real normed linear space X is k-rotund if and only if for any linearly independent x1,…,xk+1∈X, ‖x1+⋯+xk+1‖<‖x1‖+⋯+‖xk+1‖. We show that X is k-uniformly rotund if and only if the set {∑i=1k+1‖xi‖−‖∑i=1k+1xi‖:x1,…,xk+1∈BX,V(x1,…,xk+1,0)≥ϵ} is bounded below by a positive number for every positive ϵ. Using these characterizations of k-rotundity and k-uniform rotundity, we obtain necessary and sufficient conditions for a space in a class of direct sums of finitely many normed spaces, to be n-rotund and n-uniformly rotund respectively for some n∈N. Moreover, we prove that, given n∈N, the 1-direct sum of normed spaces X1,…,Xd is n-uniformly rotund if and only if there exist ki∈N such that Xi is ki-uniformly rotund for i∈{1,…,d} and n=k1+⋯+kd.