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Hyperconvexity in partial metric spaces: challenges and outlooks

Dariusz Bugajewski, Piotr Kasprzak, Olivier Olela-Otafudu

TL;DR

This work probes how hyperconvexity can be formulated in partial metric spaces, introducing AP-hyperconvex and nodally hyperconvex notions and comparing them with $p^m$-hyperconvex, $d_m$-hyperconvex, and $D$-hyperconvex frameworks. It shows that naive metric analogies do not preserve key properties such as completeness and total convexity, and it analyzes several Lipschitz notions in partial metric spaces, proposing a robust contraction-like definition that fixes some shortcomings while illustrating that fixed-point results may still fail in this setting. The results map out a nuanced landscape of hyperconvexity in partial metric spaces and identify several open questions about implications between notions and extensions, with implications for fixed-point theory and extensions of Lipschitz maps in partially metric contexts.

Abstract

In this article, we present several different ways to define hyperconvexity in partial metric spaces. In particular, we show that the analogue of the Aronszajn--Panitchpakdi notion of hyperconvexity fails to exhibit certain key properties present in the classical metric setting.

Hyperconvexity in partial metric spaces: challenges and outlooks

TL;DR

This work probes how hyperconvexity can be formulated in partial metric spaces, introducing AP-hyperconvex and nodally hyperconvex notions and comparing them with -hyperconvex, -hyperconvex, and -hyperconvex frameworks. It shows that naive metric analogies do not preserve key properties such as completeness and total convexity, and it analyzes several Lipschitz notions in partial metric spaces, proposing a robust contraction-like definition that fixes some shortcomings while illustrating that fixed-point results may still fail in this setting. The results map out a nuanced landscape of hyperconvexity in partial metric spaces and identify several open questions about implications between notions and extensions, with implications for fixed-point theory and extensions of Lipschitz maps in partially metric contexts.

Abstract

In this article, we present several different ways to define hyperconvexity in partial metric spaces. In particular, we show that the analogue of the Aronszajn--Panitchpakdi notion of hyperconvexity fails to exhibit certain key properties present in the classical metric setting.
Paper Structure (5 sections, 6 theorems, 39 equations)

This paper contains 5 sections, 6 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hyperconvexity
  4. Lipschitz continuity
  5. Conclusions

Key Result

Proposition 2

Let $(U,p)$ be a partial metric space consisting of at least two points. Assume that for any pair of closed balls ${\overline{B}}_p(x,r)$ and ${\overline{B}}_p(y,R)$ whose centers satisfy the condition $p(x,y)\leq r+R$, there exists a point $z \in U$ such that $p(x,z)\leq r$ and $p(z,y) \leq R$. The

Theorems & Definitions (29)

  • Definition 1
  • Proposition 2
  • proof
  • Definition 3
  • Remark 4
  • Definition 5
  • Remark 6
  • Example 7
  • Example 8
  • Remark 9
  • ...and 19 more