ON RELATIVE k-UNIFORM ROTUNDITY, NORMAL STRUCTURE AND FIXED POINT PROPERTY FOR NONEXPANSIVE MAPS

摘要

The idea of k-uniform rotundity relative to a k-dimensional subspace generalizes the classical notion of uniform rotundity in a direction. A normed space that is k-uniformly rotund relative to every k-dimensional subspace is said to be UREk. In this article, relative k-uniform rotundity is used to obtain: (1) new conditions sufficient for asymptotic centers to be compact; (2) new equivalent conditions for normal structure, weak normal structure and weak fixed point property for nonexpansive maps (WFPP) in a normed space. These results are then applied to study the inheritance of some geometric and fixed point properties in products of normed spaces. In particular, it is proved that if N-1, N-2 are normed spaces such that N-1 is UREk1 and N-2 is UREk2, then for 1 < p < infinity, the p direct sum of N-1 and N-2 is URE(k1+k2-1). Also, it is proved that if N-1 has WFPP and N-2 is UREk for some positive integer k, then with respect to certain norms (including the standard p-norms for 1 < p < infinity) N-1 circle times N-2 has WFPP. In addition to these, relative k-uniform rotundity in a class of subspaces of the Banach space of all bounded real valued functions on a nonempty set with supremum norm is also studied.