A note on remotal and uniquely remotal sets in normed linear spaces
Uddalak Mukherjee, Sumit Som
TL;DR
The paper addresses the problem of identifying remotal and uniquely remotal subsets in real normed spaces using generalized convergence notions. It develops $\alpha\beta$-statistical convergence tools and introduces $x$-$\alpha\beta$-compactness and partial $x$-$\alpha\beta$-compactness, proving that $x$-$\alpha\beta$-compactness implies attainment of the farthest distance $\delta(x,E)$ and hence remotality from $x$, with stronger conclusions when this holds for all $x$. A key contribution is formalizing a weakening of $x$-compactness that still guarantees farthest-point attainment, relating to max-Chebyshev points, and clarifying the relation to classical compactness via examples. This framework broadens sufficient conditions for the farthest point problem in Banach spaces by leveraging $\alpha\beta$-statistical convergence to establish existence and potential uniqueness of farthest points.
Abstract
Remotal and uniquely remotal sets play an important role in the area of farthest point problem as well as nearest point problem in a Banach space $X.$ In this study, we find some sufficient conditions for remotality and uniquely remotality of a bounded subset of a Banach space $X$ through $αβ$-statistical convergence.