Distance Equilibrium Measures and Curvature in Metric Spaces

TL;DR

This paper explores measures in metric spaces that reflect a curvature-like property, showing their existence relates to the curvature of curves and extending the concept beyond smooth settings.

Contribution

It introduces a new integral-based notion of curvature in metric spaces and analyzes its existence in relation to the geometric properties of curves.

Findings

Measures exist for curves with near-constant curvature

Single small-curvature points prevent such measures from existing

Connections to graph curvature and magnitude are discussed

Abstract

Let $(X,d)$ be a compact metric space. We consider the behavior of probability measures $\mu$ with the property that $$ \int_{X} d(x, y) d\mu(y) \qquad \mbox{is independent of}~x \in X.$$ It appears that such measures, when they exist, encode a `curvature-type' quantity. We investigate this in the special case where $X$ is a closed, convex curve in $\mathbb{R}^2$ and $d = \| \cdot \|_2$ is the Euclidean distance: even a single point with small curvature implies non-existence of such a measure. Conversely, such a measure $\mu$ exists for all curves whose curvature is sufficiently close to constant. Curvature is usually defined by second derivatives; this one is defined via an integral equation which makes sense in much rougher spaces. Connections to curvature on graphs, the Gross-Stadje Theorem and magnitude are discussed.