Distance Equilibrium Measures and Curvature in Metric Spaces

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 $\mathbb{R}<sup>2$ 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.

• The paper introduces distance equilibrium measures whose constant expected-distance equation provides a curvature-like invariant for nonsmooth metric spaces, extending beyond differential geometry.
• The paper proves structural non-existence for certain nearly flat convex curves and establishes existence for plane curves with sufficiently near-constant curvature, while identifying unresolved threshold behavior.
• The paper shows curvature measures approximately satisfy the equilibrium equation and that discrete solutions can become signed measures reflecting positive and negative curvature on non-convex curves.

The distance equilibrium equation

Let $(X,d)$ be a compact metric space. The paper studies probability measures $\mu$ on $X$ satisfying the integral equation

$$\int_X d(x,y)\, d\mu(y) = c \quad \text{for all } x \in X,$$

i.e., the expected distance from $x$ to a $\mu$-random point is independent of $x$. Such measures are called distance equilibrium measures. The central proposal of the paper is that, when such a measure exists, it encodes a curvature-type quantity for $(X,d)$ — one defined by an integral equation rather than by differentiation, and therefore meaningful in spaces far rougher than smooth manifolds.

Two canonical examples anchor the intuition: on $[0,1]$, the measure placing mass $1/2$ at each endpoint solves the equation (endpoints as "infinite curvature," interior flat), while on $\mu$0 the normalized surface measure is a solution, consistent with constant curvature. Existence is delicate: the distance kernel defines a compact operator, so solvability fails for most domains.

Structural obstructions

A result due to Wilson (summarized from Cleary–Morris–Yost) sharply constrains which subsets of $\mu$1 can admit an equilibrium measure: if $\mu$2 is compact, connected, and admits such a $\mu$3, then $\mu$4 is either a line segment or contains no three collinear points. The proof is elementary — strict convexity of the norm forces a strict inequality at any interior collinear point unless all mass sits at the segment endpoints. This immediately identifies strictly convex closed curves in $\mu$5 as the natural class to study.

For that class, the paper proves a non-existence result with explicit constants: if $\mu$6 is a smooth, closed, strictly convex curve contained in the annulus $\mu$7 and possesses even a single point with curvature $\mu$8, then no probability measure solves the equation. The argument is a second-difference estimate: three nearly collinear points $\mu$9, $X$0, $X$1 on the curve yield, via convexity of $X$2, a lower bound on the signed combination of distances that must vanish if the integral were constant; taking $X$3 forces $X$4, while a separate fluctuation argument gives $X$5. Hence $X$6, i.e., curvature at least $X$7 everywhere. The author notes the constants are not optimized and that the annulus assumption is presumably unnecessary but simplifies the proof; the argument extends formally to higher dimensions, though quantitative critical-curvature estimates are already challenging in dimension two.

Existence for nearly round curves

The main theorem asserts existence under near-constant curvature: there is a universal $X$8 such that every smooth, closed, strictly convex plane curve satisfying

$X$9

admits an equilibrium probability measure. The proof proceeds by contradiction over a sequence of counterexamples $$\int_X d(x,y)\, d\mu(y) = c \quad \text{for all } x \in X,$$0 normalized to length $$\int_X d(x,y)\, d\mu(y) = c \quad \text{for all } x \in X,$$1. A quantitative version of the Fundamental Theorem of Plane Curves shows such curves converge uniformly (after rigid motion) to the unit circle at rate $$\int_X d(x,y)\, d\mu(y) = c \quad \text{for all } x \in X,$$2. For each enclosed domain $$\int_X d(x,y)\, d\mu(y) = c \quad \text{for all } x \in X,$$3, Björck's 1958 theorem supplies a unique maximizer $$\int_X d(x,y)\, d\mu(y) = c \quad \text{for all } x \in X,$$4 of the energy $$\int_X d(x,y)\, d\mu(y) = c \quad \text{for all } x \in X,$$5, supported on $$\int_X d(x,y)\, d\mu(y) = c \quad \text{for all } x \in X,$$6 and satisfying the Euler–Lagrange condition that $$\int_X d(x,y)\, d\mu(y) = c \quad \text{for all } x \in X,$$7 equals $$\int_X d(x,y)\, d\mu(y) = c \quad \text{for all } x \in X,$$8 on the support and is bounded above elsewhere. If the support were dense in the boundary, continuity would give the desired solution; otherwise there is an unsupported arc between two support points $$\int_X d(x,y)\, d\mu(y) = c \quad \text{for all } x \in X,$$9, $x$0, and the Euler–Lagrange inequalities force a signed integral inequality ($x$1) that must be contradicted.

Two cases arise. When the arc endpoints stay at positive scale ($x$2), Gromov–Hausdorff convergence to a circle makes the integrand explicitly computable, with only two roots, and positivity on the support contradicts ($x$3). When $x$4, the integrand degenerates at scale $x$5; the paper derives a lower bound $x$6 from the curvature lower bound, obtains an asymptotic expansion of the integrand valid away from the origin (positive outside a disk of radius $x$7 centered at $x$8), and controls the near-origin contribution via monotonicity properties of a simplified function $x$9 together with a Taylor expansion along the curve. A closing remark identifies, via Taylor expansion, a critical threshold $\mu$0 (curvature $\mu$1) in the local model, suggesting existence and non-existence are governed by sharp curvature thresholds. The author concedes the method is far from optimal quantitatively and is currently the only known route to existence.

Curvature as an approximate solution

Beyond existence, the equilibrium density itself tracks curvature. Using the support-function parametrization $\mu$2, where arclength is $\mu$3 and curvature is $\mu$4, the paper shows that the curvature measure $\mu$5 nearly solves the equation: the oscillation

$\mu$6

satisfies $\mu$7. Notably, the error depends only on the deviation of $\mu$8 from a constant and its first derivative, whereas $\mu$9 itself involves $x$0 — a cancellation between curvature and the arclength element drives the result. Numerical experiments confirm the rescaled equilibrium density closely matches curvature as a function of arclength, though the author states Proposition 3 may not capture the full picture.

Discretization and signed measures

Although the continuous equation may have no solution, its discretization typically does. Sampling a curve $x$1 at $x$2 points with arclength weights yields a linear system $x$3 with $x$4; invertibility of Euclidean distance matrices (Micchelli) supports an existence theory for curves. Strikingly, for non-convex curves — where no probability solution can exist by Wilson's obstruction — the discrete solution is a signed measure that is positive on convex portions and negative on concave ones, mirroring classical signed curvature. Discretized planar domains behave similarly: solutions concentrate strongly at corners and identify boundary geometry, including opening angles of a triangle, apparently converging weakly to singular signed measures. The surface case, where local Voronoi volumes enter, is identified as more delicate.

Related frameworks

The equation connects to several established lines of work:

• Graph curvature: Steinerberger's graph curvature solves the analogous linear system with the graph distance matrix, and has been developed by Chen–Tsui, Cushing et al., Robertson, and others; resistance distance yields Devriendt–Ottolini–Steinerberger's resistance curvature. Unlike Ollivier–Ricci or Lin–Lu–Yau curvature, this notion is global yet empirically captures local structure; the present paper is motivated by finding its continuous analogue.
• Gross–Stadje theory: Gross's rendezvous number $x$5 — the unique value realized as an average distance from some point for every finite sample — equals the constant $x$6 whenever an equilibrium measure exists, providing one of the few exact computation methods.
• Distance geometry: Maximizing $x$7 relates to Alexander–Stolarsky energy problems, transfinite diameter (Fekete–Szegő), Grove–Markvorsen rigidity in Alexandrov spaces, and Kokkendorff's characterization of the sphere among Ricci-low manifolds.
• Magnitude: Leinster's weight measure satisfies the same equation with kernel $x$8; magnitude is a single invariant, whereas the paper concerns the measure itself. The precise relationship among these notions remains unclear.

Limitations and open questions

Several limitations are stated plainly. The existence constant $x$9 is unspecified and, if made explicit, would likely be far from optimal; the gap between the non-existence threshold (curvature $(X,d)$0 locally kills existence) and the existence regime is not understood. The annulus hypothesis in the non-existence proposition is not believed necessary. The higher-dimensional boundary problem — where Gaussian curvature should govern — is entirely open, as is a complete existence theory for plane curves. Signed-measure formulations, which appear numerically to encode both positive and negative curvature, lack any rigorous theory. Finally, the discretized systems always seem solvable and geometrically informative even when the limiting equation has no solution, and the sense in which these discrete solutions converge to singular signed measures is unproven.

Conclusion

The paper establishes that the distance equilibrium equation, long studied as a computational device for Gross's rendezvous number, carries genuine geometric content: on plane convex curves, existence is controlled by curvature through matching upper and lower thresholds, the equilibrium density approximates curvature with an error depending only on first-order support-function data, and discretizations produce signed measures reflecting signed curvature even where continuous solutions cannot exist. The main open problems — sharp existence criteria, the Gaussian-curvature analogue in higher dimensions, and a convergence theory for the discrete signed solutions — define a concrete agenda connecting integral equations, distance geometry, and metric curvature.