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Distance Equilibrium Measures and Curvature in Metric Spaces

Stefan Steinerberger

TL;DR

The paper introduces a distance-based integral equation, $f(x)=\int_{X} d(x,y)\,d\mu(y)$, as a curvature-type quantity on compact metric spaces and focuses on $X$ being a smooth, closed convex curve in the plane with Euclidean distance. It establishes a nonexistence result for curves with a single very small curvature point, and proves an existence result when the curvature is nearly constant via a Björck-type maximizing-measure argument, connecting the continuum problem to boundary-supported measures. It also situates the framework within graph curvature, the Gross-Stadje theorem, differential geometry, and Leinster's magnitude, and analyzes the curvature as an almost-solution in a near-round limit, with discretizations yielding informative linear systems. Together, these results provide a novel, integral-equation-based perspective on curvature that extends to rough spaces and links discrete and continuous curvature notions. The insights offer potential avenues for defining and analyzing curvature-like quantities on general metric spaces, graphs, and convex boundaries beyond classical differential geometry.

Abstract

Let $(X,d)$ be a compact metric space. We consider the behavior of probability measures $μ$ with the property that $$ \int_{X} d(x, y) dμ(y) \qquad \mbox{is independent of}~x \in X.$$ 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 $μ$ 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.

Distance Equilibrium Measures and Curvature in Metric Spaces

TL;DR

The paper introduces a distance-based integral equation, , as a curvature-type quantity on compact metric spaces and focuses on being a smooth, closed convex curve in the plane with Euclidean distance. It establishes a nonexistence result for curves with a single very small curvature point, and proves an existence result when the curvature is nearly constant via a Björck-type maximizing-measure argument, connecting the continuum problem to boundary-supported measures. It also situates the framework within graph curvature, the Gross-Stadje theorem, differential geometry, and Leinster's magnitude, and analyzes the curvature as an almost-solution in a near-round limit, with discretizations yielding informative linear systems. Together, these results provide a novel, integral-equation-based perspective on curvature that extends to rough spaces and links discrete and continuous curvature notions. The insights offer potential avenues for defining and analyzing curvature-like quantities on general metric spaces, graphs, and convex boundaries beyond classical differential geometry.

Abstract

Let be a compact metric space. We consider the behavior of probability measures with the property that It appears that such measures, when they exist, encode a `curvature-type' quantity. We investigate this in the special case where is a closed, convex curve in and is the Euclidean distance: even a single point with small curvature implies non-existence of such a measure. Conversely, such a measure 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.
Paper Structure (18 sections, 8 theorems, 76 equations, 13 figures)

This paper contains 18 sections, 8 theorems, 76 equations, 13 figures.

Table of Contents

  1. Introduction and Results
  2. Introduction.
  3. Small curvature implies non-existence.
  4. An Existence Result
  5. Open problems.
  6. Related topics and ideas
  7. Graph curvature.
  8. Gross-Stadje Theorem.
  9. Differential Geometry.
  10. Magnitude.
  11. Proof of Proposition 1 and Proposition 2
  12. Proof of Proposition 1
  13. Proof of Proposition 2
  14. Proof of the Theorem
  15. Main Idea.
  16. ...and 3 more sections

Key Result

Proposition 1

Let $X \subset \mathbb{R}^n$ be compact and connected. If there exists a probability measure $\mu$ on $X$ such that then $X$ is either a line segment or does not contain three points on any line.

Figures (13)

  • Figure 1: A measure $\mu$ on the unit interval $X = [0,1]$ (left) and the sphere (right) solving the integral equation.
  • Figure 2: Left: a convex curve $X \subset \mathbb{R}^2$. Middle: an approximation of the density of $\mu$ as a function of the arclength. Right: the curve colored by the density of $\mu$ (red means higher density).
  • Figure 3: Curves colored by the density of $\mu$ (red means higher density).
  • Figure 4: A non-convex curve colored by density of the (signed) measure (left) and the density of the signed measure written as the numerical solution of a discretization (right).
  • Figure 5: Numerical solutions of a discretized square (left) and a discretized right-angled triangle (right). The numerical solutions are large and positive in corners, slightly less large at the boundary.
  • ...and 8 more figures

Theorems & Definitions (14)

  • Proposition 1: Wilson cleary
  • Proposition 2
  • Theorem
  • Theorem : Gross, 1964
  • proof : Proof summarized from cleary
  • proof
  • Theorem : Björck bjorck
  • proof : Sketch of the Proof.
  • Corollary 1
  • proof
  • ...and 4 more