Characterizations of some rotund properties in terms of farthest points

Abstract

We characterize rotund, uniformly rotund, locally uniformly rotund and compactly locally uniformly rotund spaces in terms of set of almost farthest points from the unit sphere using the generalized diameter. For this we introduce few notions involving the almost farthest points, namely strongly remotal, strongly uniquely remotal and uniformly strongly uniquely remotal sets. As a consequence, we obtain some characterizations of the aforementioned rotundity properties in terms of existing proximinality notions.