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Some reverse inequality in optimal mass transportation

Luigi De Pascale, Igor Pinheiro

TL;DR

This work develops a general framework to bound the supremal transport cost $\mathcal{C}_\infty(\rho)$ from below by a function of the Kantorovich-type cost $\mathcal{C}(\rho)$ for repulsive, pointwise transport costs $c(x_1,\dots,x_N)=\sum_{i<j} h(d(x_i,x_j))$ on a Polish space. The main result provides an explicit lower bound that depends on the concentration function $\kappa_\rho$ and the inverse of $h$, yielding a versatile reverse inequality that subsumes Wasserstein-type and Coulomb-type costs. The paper also specializes to the two-marginal case with sharper Fréchet-type bounds and derives concrete lower bounds for several measure classes, including unimodal isotropic densities and discrete measures. These results offer a unifying, quantitative tool for lower-bounding multimarginal transport costs under regularity or structural assumptions on the marginals, with potential implications for PDEs, regularity theory, and numerical methods. The approach highlights how concentration and tail behavior of the marginals control the gap between $\mathcal{C}$ and $\mathcal{C}_\infty$, and identifies practical classes where explicit constants can be obtained (e.g., Gaussian radially symmetric densities and discrete atom configurations).

Abstract

Controlling the $\mathcal W_\infty$ Wasserstein distance by the $\mathcal W_p$ Wasserstein distance is interesting both for theorical and numerical applications. A first paper on this problem was written several years ago [3]. Some year later [14] framed it in the same inequality for more general costs which increase with the distance. In this paper, we prove this type of inequality for optimal transport problems with pointwise cost which is a decreasing function of the distance. We show, in particular, that there is a general framework that encompasses all the cases above.

Some reverse inequality in optimal mass transportation

TL;DR

This work develops a general framework to bound the supremal transport cost from below by a function of the Kantorovich-type cost for repulsive, pointwise transport costs on a Polish space. The main result provides an explicit lower bound that depends on the concentration function and the inverse of , yielding a versatile reverse inequality that subsumes Wasserstein-type and Coulomb-type costs. The paper also specializes to the two-marginal case with sharper Fréchet-type bounds and derives concrete lower bounds for several measure classes, including unimodal isotropic densities and discrete measures. These results offer a unifying, quantitative tool for lower-bounding multimarginal transport costs under regularity or structural assumptions on the marginals, with potential implications for PDEs, regularity theory, and numerical methods. The approach highlights how concentration and tail behavior of the marginals control the gap between and , and identifies practical classes where explicit constants can be obtained (e.g., Gaussian radially symmetric densities and discrete atom configurations).

Abstract

Controlling the Wasserstein distance by the Wasserstein distance is interesting both for theorical and numerical applications. A first paper on this problem was written several years ago [3]. Some year later [14] framed it in the same inequality for more general costs which increase with the distance. In this paper, we prove this type of inequality for optimal transport problems with pointwise cost which is a decreasing function of the distance. We show, in particular, that there is a general framework that encompasses all the cases above.
Paper Structure (7 sections, 20 theorems, 105 equations)

This paper contains 7 sections, 20 theorems, 105 equations.

Table of Contents

  1. Description of the problem and of the results
  2. The case of Wassertein distances
  3. Main theorem
  4. The case of two marginals
  5. Classes of probability measures
  6. Unimodal isotropic probability measures
  7. The case of discrete measures

Key Result

Theorem 1.5

Let $(X,d)$ be a complete metric space, $h:[0,+\infty) \to [0,+\infty)$ a non-decreasing function with $h(t)>0$ for all $t>0$ and let $\mu \in \mathcal{P}(X)$. Then, there exists a non-decreasing function $\omega:[0, +\infty) \to [0, +\infty)$ with $\omega(t)>0$ for all $t>0$ such that if and only if $spt(\mu)$ is compact and connected. Moreover, in such case one can take $\omega(t)= \frac{1}{2}m

Theorems & Definitions (61)

  • Definition 1.1
  • Remark 1.2
  • Example 1.3
  • Example 1.4
  • Theorem 1.5: Th. 1.1, JylRaj2016JFA
  • Remark 1.6
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • ...and 51 more