Farthest Point Problem and Partial Statistical Continuity in Normed Linear Spaces

TL;DR

This paper investigates conditions under which uniquely remotal subsets in normed linear spaces are singletons, focusing on partial statistical continuity of the farthest point map and its implications in various Banach spaces.

Contribution

It establishes that certain partial statistical continuity conditions imply a uniquely remotal set must be a singleton, and explores examples where continuity fails but partial statistical continuity holds.

Findings

Uniquely remotal sets with a Chebyshev center are singletons under partial statistical continuity.

Necessary conditions for remotal sets in uniformly rotund Banach spaces to be singletons.

Existence of remotal sets with partial statistical continuity but not full continuity of the farthest point map.

Abstract

In this paper, we prove that if $E$ is a uniquely remotal subset of a real normed linear space $X$ such that $E$ has a Chebyshev center $c \in X$ and the farthest point map $F:X\rightarrow E$ restricted to $[c,F(c)]$ is partially statistically continuous at $c$, then $E$ is a singleton. We obtain a necessary condition on uniquely remotal subsets of uniformly rotund Banach spaces to be a singleton. Moreover, we show that there exists a remotal set $M$ having a Chebyshev center $c$ such that the farthest point map $F:\mathbb{R}\rightarrow M$ is not continuous at $c$ but is partially statistically continuous there in the multivalued sense.