Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The local diameter two property and the diameter two property in spaces of Lipschitz functions

Rainis Haller, Jaan Kristjan Kaasik, Andre Ostrak

TL;DR

This work disentangles the local and global diameter-two properties and their weak*-counterparts within spaces of Lipschitz functions. By developing a dual representation of Lip$_0(M)^*$ via a de Leeuw-type construction and introducing a generalized $\gamma$-cyclic monotonicity, it provides precise dual and CM-based criteria for LD$2$P, D$2$P, SD$2$P, and their $w^*$-variants. The authors furnish metric characterizations such as Lip-LTP and 2-Lip-LTP that determine when Lip$_0(M)$ has the $w^*$-D$2$P or the $w^*$-LD$2$P, and prove that these properties are distinct by constructing a metric space $M$ for which Lip$_0(M)$ has LD$2$P but not $w^*$-D$2$P. The results sharpen the understanding of octahedrality-like properties in Lip$_0(M)$ and supply tools that may be of independent interest for studying dual representations and diameter-two phenomena in Lipschitz-function spaces.

Abstract

We separate the local diameter two property from the diameter two property and their weak-star counterparts from each other in spaces of Lipschitz functions. We characterise the $w^*$-LD$2$P, the $w^*$-D$2$P, the LD$2$P, and the SD$2$P in these spaces. We introduce a generalised version of cyclical monotonicity to study functionals on spaces of Lipschitz functions.

The local diameter two property and the diameter two property in spaces of Lipschitz functions

TL;DR

This work disentangles the local and global diameter-two properties and their weak*-counterparts within spaces of Lipschitz functions. By developing a dual representation of Lip via a de Leeuw-type construction and introducing a generalized -cyclic monotonicity, it provides precise dual and CM-based criteria for LDP, DP, SDP, and their -variants. The authors furnish metric characterizations such as Lip-LTP and 2-Lip-LTP that determine when Lip has the -DP or the -LDP, and prove that these properties are distinct by constructing a metric space for which Lip has LDP but not -DP. The results sharpen the understanding of octahedrality-like properties in Lip and supply tools that may be of independent interest for studying dual representations and diameter-two phenomena in Lipschitz-function spaces.

Abstract

We separate the local diameter two property from the diameter two property and their weak-star counterparts from each other in spaces of Lipschitz functions. We characterise the -LDP, the -DP, the LDP, and the SDP in these spaces. We introduce a generalised version of cyclical monotonicity to study functionals on spaces of Lipschitz functions.
Paper Structure (5 sections, 16 theorems, 79 equations, 1 figure)

This paper contains 5 sections, 16 theorems, 79 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Dual of the space of Lipschitz functions
  3. Characterisation of diameter 2 properties in Lip0(M)
  4. Metric characterisation of w*-D2P and w*-LD2P for spaces of Lipschitz functions
  5. Separating LD2P and D2P in spaces of Lipschitz functions

Key Result

Proposition 2.2

Let $A$ be a subset of $\widetilde{M}$ and let $\gamma\in(0,1]$. Then $A$ is $\gamma$-cyclically monotonic if and only if there exists $f\in B_{\mathop{\mathrm{Lip}}\nolimits_0(M)}$ satisfying $f(m_{x,y})\geq \gamma$ for all $(x,y)\in A$.

Figures (1)

  • Figure 1: A representation of the metric space M in Example \ref{['Ex:new']}. The distances between points connected by a straight line segment are 1, the distances between other different points are 2.

Theorems & Definitions (35)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • proof
  • Proposition 2.4
  • proof
  • Definition 2.5
  • Corollary 2.6
  • Lemma 2.7
  • ...and 25 more