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Chebyshev centers that are not farthest points

Debmalya Sain, Vladimir Kadets, Kallol Paul, Anubhab Ray

Abstract

In this paper we address the question whether in a given Banach space, a Chebyshev center of a nonempty bounded subset can be a farthest point of the set. Our exploration reveals that the answer depends on the convexity properties of the Banach space. We obtain a characterization of two-dimensional real strictly convex spaces in terms of Chebyshev center not contributing to the set of farthest points. We explore the scenario in uniformly convex Banach spaces and further study the roles played by centerability and M-compactness in the scheme of things to obtain a step by step characterization of strictly convex Banach spaces. We also illustrate with examples the optimality of our results.

Chebyshev centers that are not farthest points

Abstract

In this paper we address the question whether in a given Banach space, a Chebyshev center of a nonempty bounded subset can be a farthest point of the set. Our exploration reveals that the answer depends on the convexity properties of the Banach space. We obtain a characterization of two-dimensional real strictly convex spaces in terms of Chebyshev center not contributing to the set of farthest points. We explore the scenario in uniformly convex Banach spaces and further study the roles played by centerability and M-compactness in the scheme of things to obtain a step by step characterization of strictly convex Banach spaces. We also illustrate with examples the optimality of our results.

Paper Structure

This paper contains 3 sections, 16 theorems, 51 equations, 2 figures.

Table of Contents

  1. Introduction.
  2. Main results
  3. Chebyshev centers in $L_p$ spaces

Key Result

Lemma 2.1

Let $A$ be a nontrivial bounded subset of $X$, $x \in \mathop{\rm Far}\nolimits{A}$. Then, for every $N > 0$ there is a point $y \in X$ such that $x$ is a farthest point of $A$ from $y$ and $\| x-y \| > N$.

Figures (2)

  • Figure :
  • Figure :

Theorems & Definitions (35)

  • Definition 1.1
  • Definition 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 25 more