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Farthest point problem and M-compact sets

D. Sain, K. Paul, A. Ray

Abstract

In this paper we give an elementary proof of the fact that every uniquely remotal set is singleton in a finite dimensional strictly convex normed linear space. We show that if A is a uniquely remotal M-compact subset with the derived set of A non-empty then the derived set of A is M-compact and uniquely remotal. We also show that if A is a uniquely remotal M-compact set and the derived set of A is compact then A is singleton.

Farthest point problem and M-compact sets

Abstract

In this paper we give an elementary proof of the fact that every uniquely remotal set is singleton in a finite dimensional strictly convex normed linear space. We show that if A is a uniquely remotal M-compact subset with the derived set of A non-empty then the derived set of A is M-compact and uniquely remotal. We also show that if A is a uniquely remotal M-compact set and the derived set of A is compact then A is singleton.

Paper Structure

This paper contains 2 sections, 8 theorems, 7 equations.

Table of Contents

  1. Introduction.
  2. Main Results.

Key Result

Theorem 2.1

Let $X$ be a strictly convex normed linear space. If $A \subseteq X$ is uniquely remotal then $\overline{A}$ is also uniquely remotal.

Theorems & Definitions (16)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • proof
  • ...and 6 more