Farthest Point Problem and Partial Statistical Continuity in Normed Linear Spaces

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 $F:X\to E$ 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 $F:\mathbb{R}\to M$ is not continuous at $c$ but is partially statistically continuous there in the multivalued sense.

arXiv comment8 pages

References in corpus (2)

• On remotality for convex sets in Banach spaces
• On the farthest point problem in Banach spaces

Cited by in corpus (1)

• Characterization of $M$-compact sets via statistically convergent sequences