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

In this study, we conduct a literature review on normed linear spaces whose strengths are between rotundity and uniform rotundity. In this discourse, we also explore inter-relationships and juxtapositions between the subjects under consideration. There has been some discussion on the extent to which the geometry of the factor spaces has an impact on the geometry of the product spaces, as well as the degree to which the quotient spaces and subspaces inherit the geometry of the space itself. A comprehensive review has been conducted on the applications of most of these rotundities to some fields within the realm of approximation theory. In addition, some open problems are enumerated in the paper.

We introduce two types of mappings, namely Reich type nonexpansive and Chatterjea type nonexpansive mappings, and derive some sufficient conditions under which these two types of mappings possess an approximate fixed point sequence (AFPS). We obtain the desired AFPS using the well-known Scha¨efer iteration method. Along with these, we check some special properties of the fixed point sets of these mappings, such as closedness, convexity, remotality, unique remotality, etc. We also derive a nice interrelation between AFPS and maximizing sequence for both types of mappings. Finally, we will get some sufficient conditions under which the class of Reich type nonexpansive mappings reduces to that of nonexpansive maps.

A Chebyshev set is a subset of a normed linear space that admits unique best approximations. In the first part of this paper we present some basic results concerning Chebyshev sets. In particular, we investigate properties of the metric projection map, sufficient conditions for a subset of a normed linear space to be a Chebyshev set, and sufficient conditions for a Chebyshev set to be convex. In the second half of the paper we present a construction of a nonconvex Chebyshev subset of an inner product space.

Farthest point theory is not so rich and developed as nearest point theory, which has more applications. Farthest points are useful in studying the extremal structure of sets; see, e.g., the survey paper [14]. There are some interactions between the two theories; in particular, uniquely remotal sets in Hilbert spaces are related to the old open problem concerning the convexity of Chebyshev sets. The aim of this paper is twofold: first, we indicate characterizations of inner product spaces and of infinite-dimensional Banach spaces, in terms of remotal points and uniquely remotal sets. Second, we try to update the survey paper [15], concerning uniquely remotal sets.

The study of the continuity of the farthest point mapping for uniquely remotal sets has been used extensively in the literature to prove the singletoness of such sets. In this article, we show that the farthest point mapping is not continuous even if the set is remotal, rather than being uniquely remotal. Consequently, we obtain some generalizations of results concerning the singletoness of remotal sets. In particular, it is proved that a compact set admitting a unique farthest point to its center is a singleton, generalizing the well known result of Klee.