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Non-uniformly continuous nearest point maps

Rubén Medina, Andrés Quilis

Abstract

We construct a Banach space satisfying that the nearest point map (also called proximity mapping or metric projection) onto any compact and convex subset is continuous but not uniformly continuous. The space we construct is locally uniformly convex, which ensures the continuity of all these nearest point maps. Moreover, we prove that every infinite-dimensional separable Banach space is arbitrarily close (in the Banach-Mazur distance) to one satisfying the above conditions.

Non-uniformly continuous nearest point maps

Abstract

We construct a Banach space satisfying that the nearest point map (also called proximity mapping or metric projection) onto any compact and convex subset is continuous but not uniformly continuous. The space we construct is locally uniformly convex, which ensures the continuity of all these nearest point maps. Moreover, we prove that every infinite-dimensional separable Banach space is arbitrarily close (in the Banach-Mazur distance) to one satisfying the above conditions.
Paper Structure (6 sections, 4 theorems, 17 equations, 1 figure)

This paper contains 6 sections, 4 theorems, 17 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main result and background
  3. Outline
  4. Notation and preliminaries
  5. Blueprint norm
  6. Proof of main theorem

Key Result

Lemma 2.2

Let $\alpha=(v,v^*,e,e^*,h,h^*,t)$ as above. Let $\lambda=1-\frac{\rho}{100}\in \left(\frac{1}{2},1\right)$, and consider: Let $k\in B_{(X,\|\cdot\|)}$. Then, we have: Similarly, we also obtain:

Figures (1)

  • Figure 1: Top half of the unit ball of $\|\cdot\|_\alpha$ in the three-dimensional euclidean space using the vectors from the canonical biorthogonal basis.

Theorems & Definitions (10)

  • Remark 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Corollary 3.3: Theorem \ref{['THEOREM_A']}
  • proof