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arxiv: 1907.09023 · v1 · pith:56LC5JGVnew · submitted 2019-07-21 · 🧮 math.CA · math-ph· math.MP

A Wasserstein Inequality and Minimal Green Energy on Compact Manifolds

Stefan Steinerberger This is my paper

Pith reviewed 2026-05-24 18:09 UTC · model grok-4.3

classification 🧮 math.CA math-phmath.MP
keywords Wasserstein distanceGreen's functiondiscrete energycompact manifoldsCoulomb kernelempirical measuresLaplacian
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The pith

An inequality bounds the Wasserstein-2 distance of n points to volume measure by the square root of their pairwise Green energy sum, implying optimal convergence for energy minimizers.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that on a smooth compact d-dimensional manifold without boundary, the Wasserstein-2 distance from the empirical measure of n points to the normalized volume measure is controlled by n to the power -1/d plus a term involving the square root of the sum of the Green's function values between distinct pairs of points. This bound is obtained by relating the transportation cost directly to the Green's function of the Laplacian, normalized to have mean zero. The same relation holds for the normalized Coulomb kernel on the sphere. As a consequence, any configuration that minimizes the discrete Green energy must satisfy the optimal rate W2 distance at most order n to the power -1/d.

Core claim

The central result states that W_2((1/n) sum delta_xk, dx) is at most a manifold-dependent constant times n^{-1/d} plus (1/n) times the square root of the absolute value of the double sum over k not equal to l of G(x_k, x_l). Minimizers of the discrete Green energy therefore obey W_2 ≲ n^{-1/d}. The identical conclusion holds for minimizers of the normalized Coulomb energy on the sphere.

What carries the argument

The inequality that expresses the Wasserstein-2 transportation cost between an empirical measure and volume measure in terms of the square root of the off-diagonal Green energy sum.

If this is right

  • Minimizers of the discrete Green energy achieve the optimal W2 convergence rate n^{-1/d} to the volume measure.
  • The same optimal rate holds for minimizers of the normalized Coulomb energy on the sphere.
  • The bound separates the geometric discrepancy term n^{-1/d} from the interaction energy term controlled by the Green's function.
  • The inequality applies uniformly with a constant depending only on the manifold.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result supplies a quantitative link between classical potential theory and optimal transport that could be used to certify good point distributions without directly computing Wasserstein distances.
  • It raises the question whether analogous inequalities exist for other kernels or for the Wasserstein-p distance with p not equal to 2.
  • On manifolds where the Green's function is explicitly known, the inequality gives a practical way to bound discretization error for integration or sampling.

Load-bearing premise

The manifold is smooth, compact, and without boundary, and the Green's function is normalized to have mean zero.

What would settle it

A sequence of n-point configurations on the manifold whose Green energy sum grows slower than n^2 while the Wasserstein-2 distance remains larger than C n^{-1/d} for any fixed C.

read the original abstract

Let $M$ be a smooth, compact $d-$dimensional manifold, $d \geq 3,$ without boundary and let $G: M \times M \rightarrow \mathbb{R} \cup \left\{\infty\right\}$ denote the Green's function of the Laplacian $-\Delta$ (normalized to have mean value 0). We prove a bound on the cost of transporting Dirac measures in $\left\{x_1, \dots, x_n\right\} \subset M$ to the normalized volume measure $dx$ in terms of the Green's function of the Laplacian $$ W_2\left( \frac{1}{n} \sum_{k=1}^{n}{\delta_{x_k}}, dx\right) \lesssim_M \frac{1}{n^{1/d}} + \frac{1}{n} \left| \sum_{k, \ell=1 \atop k \neq \ell}^{n}G(x_k, x_{\ell})\right|^{1/2}.$$ We obtain the same result for the Coulomb kernel $G(x,y) = 1/\|x-y\|^{d-2}$ on the sphere $\mathbb{S}^d$, for $d \geq 3$, where we show that $$ W_2\left(\frac{1}{n} \sum_{k=1}^{n}{ \delta_{x_k}}, dx\right) \lesssim \frac{1}{n^{1/d}} + \frac{1}{n} \left| \sum_{k, \ell=1 \atop k \neq \ell}^{n}{\left(\frac{1}{\|x_k - x_{\ell}\|^{d-2}} - c_d \right)} \right|^{\frac{1}{2}},$$ where $c_d$ is the constant that normalizes the Coulomb kernel to have mean value 0. We use this to show that minimizers of the discrete Green energy on compact manifolds have optimal rate of convergence $W_2\left( \frac{1}{n} \sum_{k=1}^{n}{\delta_{x_k}}, dx\right) \lesssim n^{-1/d}$. The second inequality implies the same result for minimizers of the Coulomb energy on $\mathbb{S}^d$ which was recently proven by Marzo & Mas.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript proves the inequality W_2((1/n) sum delta_{x_k}, dx) ≲_M n^{-1/d} + (1/n) |sum_{k≠ℓ} G(x_k, x_ℓ)|^{1/2} on a smooth compact d-manifold (d≥3) without boundary, where G is the mean-zero Green's function of -Δ. An analogous bound is established for the normalized Coulomb kernel on S^d. These are applied to show that minimizers of the discrete Green energy (and of the Coulomb energy on the sphere) achieve the optimal rate W_2 ≲ n^{-1/d}, by combining the inequality with an explicit upper bound on the minimal energy obtained from a suitable test configuration.

Significance. If the central inequality holds, the work supplies a direct, quantitative link between discrete Green/Coulomb energy and equidistribution measured in W_2 distance. The derivation uses only standard singularity and regularity properties of the Green's function together with a discrepancy-controlled test configuration; this is a clear strength. The result confirms the optimal rate for energy minimizers and recovers the recent Marzo-Mas theorem on the sphere as a special case.

minor comments (3)
  1. §1, after the statement of the main inequality: the dependence of the implicit constant on the manifold M (via its geometry and the normalization of G) is asserted but not quantified; a brief remark on the explicit dependence would improve readability.
  2. The proof of the upper bound on the minimal discrete energy (used to obtain the rate for minimizers) relies on a specific test configuration whose discrepancy is controlled at rate n^{-1/d}; a short paragraph recalling the construction and the resulting energy estimate would make the argument self-contained.
  3. In the Coulomb case on S^d, the subtracted constant c_d is defined as the mean of 1/||x-y||^{d-2}; stating its explicit value (or its integral representation) in the theorem statement would avoid cross-reference to the normalization paragraph.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments were listed in the report, so we have no points requiring point-by-point rebuttal. We will incorporate any minor suggestions during revision.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper establishes the Wasserstein bound via direct estimates on the Green's function (mean-zero normalization, |x-y|^{2-d} singularity, manifold smoothness) without any reduction to fitted parameters or self-referential definitions. The consequence for energy minimizers is obtained by pairing the inequality with an independent upper bound on minimal discrete energy from an explicit test configuration whose discrepancy is controlled at rate n^{-1/d}. The cited Marzo & Mas result is external and concerns only the sphere case; no self-citation is load-bearing, no ansatz is smuggled, and no quantity is renamed as a prediction. The derivation chain is therefore self-contained against the stated assumptions on M and G.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of the Green's function for the Laplacian on compact Riemannian manifolds (existence, symmetry, mean-zero normalization) and the definition of the 2-Wasserstein distance; no free parameters or invented entities are introduced.

axioms (2)
  • standard math Existence and basic regularity of the Green's function G for -Δ on a smooth compact manifold without boundary, normalized to mean zero.
    Invoked in the statement of the inequality; standard background from elliptic PDE theory on manifolds.
  • standard math The 2-Wasserstein distance is well-defined between the empirical measure and the normalized volume measure.
    Implicit in the left-hand side of the claimed bound.

pith-pipeline@v0.9.0 · 5967 in / 1386 out tokens · 16722 ms · 2026-05-24T18:09:41.538210+00:00 · methodology

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