Uniquely remotal sets in Banach spaces

Abstract

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.

A note on farthest point problem in Banach spaces

“…Let X be a real Banach space and A be a nonempty subset of X: The function F W A ! X is said to be partially ideal statistically continuous at a 2 X if there exists a non constant sequence fx n g n2N A such that fx n g n2N is I-statistically convergent to a and fF .x n /g n2N is I-statistically convergent to F .a/: Now we give an example to show that this notion of partial ideal statistical continuity is much weaker than partial continuity introduced by Sababheh et al [14]. Example 1.…”

“…In a natural way, in this paper we first introduce the notion of partial ideal statistical continuity of a function via ideal summability and we give an example to show that this notion of partial ideal statistical continuity is much weaker than continuity and also weaker than partial continuity introduced by Sababheh et al in [14]. We prove that if E is a non-empty, bounded, uniquely remotal subset in a real Banach space X such that E has a Chebyshev center c and the farthest point map F W X !…”

Farthest Point Problem and Partial Statistical Continuity in Normed Linear Spaces

“…Very recently Yosef, Khalil and Mutabagani [16] proved the FPP for ℓ 1 . In their work [12], Sababheh and Khalil proved that a closed and bounded set is remotal in X if and only if X is finite dimensonal. The question was answered in the negative for infinite dimensional spaces by Martin and Rao in [8], where they showed the existence of a closed bounded convex non remotal set for every infinite dimensional Banach space.…”

“…The farthest point map F : X → M is single valued only when M is uniquely remotal. In next section, we first introduce the notion of partial statistical continuity of a single valued function and provide an example to show that the notion of partial statistical continuity is much weaker than continuity as well as partial continuity introduced by Sababheh et al in [12]. We prove that if M is an uniquely remotal subset of a real normed linear space X such that M has a Chebyshev center c and the farthest point map F : X → M restricted to [c, F (c)] is partially statistically continuous at c then E is a singleton.…”

On the farthest point problem in Banach spaces

“…In [7] it was proved that if E is a uniquely remotal subset of a normed space, admitting a center c, and if F , restricted to the line segment [c, F (c)] is continuous at c, then E is a singleton. Then recently, a generalization has been obtained in [9], where the authors proved the singletoness of uniquely remotal sets if the farthest point mapping F restricted to [c, F (c)] is partially continuous at c. Furthermore, a generalization of Klee's result in [4], "If a compact subset E, with a center c, is uniquely remotal in a normed space X, then E must be a singleton", was also obtained in [9].…”