Distance Geometry in Quasihypermetric Spaces. III

TL;DR

This paper investigates the geometric constant $M(X)$ in compact metric spaces, exploring its connections to metric embeddings and properties in finite spaces, extending previous work on distance geometry in quasihypermetric spaces.

Contribution

It establishes new links between the constant $M(X)$, metric embedding properties, and the structure of finite metric spaces, advancing the understanding of distance geometry in quasihypermetric spaces.

Findings

Characterization of $M(X)$ in relation to metric embeddings.

Analysis of $M(X)$ for finite metric spaces.

Extension of previous results in distance geometry.

Abstract

Let $(X, d)$ be a compact metric space and let $\mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I \colon \mathcal{M}(X) \to \R$ by \[ I(\mu) = \int_X \int_X d(x,y) d\mu(x) d\mu(y), \] and set $M(X) = \sup I(\mu)$, where $\mu$ ranges over the collection of signed measures in $\mathcal{M}(X)$ of total mass 1. This paper, with two earlier papers [Peter Nickolas and Reinhard Wolf, Distance geometry in quasihypermetric spaces. I and II], investigates the geometric constant $M(X)$ and its relationship to the metric properties of $X$ and the functional-analytic properties of a certain subspace of $\mathcal{M}(X)$ when equipped with a natural semi-inner product. Specifically, this paper explores links between the properties of $M(X)$ and metric embeddings of $X$, and the properties of $M(X)$ when $X$ is a finite metric space.