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Some reverse inequality in optimal mass transportation

This paper establishes a general framework for proving reverse inequalities that bound the $\mathcal{W}_\infty$ Wasserstein distance by the $\mathcal{W}_p$ distance in optimal transport problems involving pointwise costs that decrease with distance, thereby unifying previous results on increasing costs.

Original authors: Luigi De Pascale, Igor Pinheiro

Published 2026-01-22

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Luigi De Pascale, Igor Pinheiro

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a logistics manager trying to move a pile of sand from one place to another. In the world of mathematics, this is called Optimal Mass Transportation. You have a starting pile of sand (a probability distribution, let's call it $\mu$ ) and a destination pile ( $\nu$ ). Your goal is to move the sand in the most efficient way possible.

Usually, "efficient" means minimizing the total amount of work done. If you move a grain of sand a short distance, it costs little. If you move it far, it costs more. The math sums up all these tiny costs to get a total price tag. This is the standard "Wasserstein distance" ( $W_p$ ).

However, sometimes you care about the worst-case scenario. You don't care about the total cost; you care about the single grain of sand that had to travel the furthest. If one grain has to travel 100 miles, your whole operation is considered "expensive," even if the other 999 grains only moved an inch. This is the "supremal" or "infinity" distance ( $W_\infty$ ).

The Big Question

The paper asks a very specific question: Can we control the "worst-case" distance using the "total" cost?

In other words, if we know the total cost of moving the sand is low, can we guarantee that no single grain of sand traveled too far?

For a long time, mathematicians knew that the total cost is always less than or equal to the worst-case cost (because the average is usually lower than the maximum). But the reverse isn't always true. You could have a tiny total cost but a massive worst-case distance if the sand is arranged in a tricky way.

The authors of this paper are trying to prove a "Reverse Inequality." They want to find a rule that says: "If the total cost is this small, then the worst-case distance cannot be bigger than that."

The Twist: Repulsive Forces

Most previous studies looked at costs that increase with distance (like paying more for a longer truck ride). This paper flips the script. They look at repulsive costs.

Imagine the sand grains are magnets with the same pole facing each other. They hate being close.

If two grains are very close, the "cost" is huge (infinite, even).
If they are far apart, the cost is tiny.

This is like the Coulomb interaction in physics (how electrons repel each other). The paper asks: If we have a bunch of these repelling particles, and we know the total repulsion energy is low, can we say anything about the maximum repulsion?

The Main Discovery

The authors found a mathematical formula that links the "Total Repulsion" to the "Maximum Repulsion."

Here is the simple analogy:
Imagine you have a crowded room of people who are all trying to stay as far apart as possible.

The "Total" view: You measure the sum of all the distances between everyone.
The "Worst" view: You look for the two people who are closest together (because that's where the repulsion is strongest).

The paper proves that if the "Total" sum of distances is small, it forces the "Worst" case (the closest pair) to be a certain distance apart.

However, there's a catch. The formula depends on how the people are distributed.

If everyone is clumped in one corner, the math breaks down (the cost becomes infinite).
If the people are spread out nicely, the formula works perfectly.

The authors introduce a concept called "Concentration." Think of this as a measure of how "clumpy" your sand or people are.

Low Concentration: The sand is spread out evenly. The formula works great.
High Concentration: The sand is in a tight pile. The formula tells us the cost might be infinite, or the relationship breaks.

The "Magic" Formula

The paper derives a specific inequality. In plain English, it says:

The Total Cost $\ge$ (A function of the Worst Cost) $\times$ (How spread out the sand is).

If the sand is very spread out (low concentration), the "Total Cost" must be significantly higher than the "Worst Cost" would suggest. If the sand is clumpy, the relationship changes.

Special Cases They Studied

The authors didn't just stop at the general rule; they looked at specific types of "sand piles" to see how the rule behaves:

The "Bell Curve" (Gaussian Distribution): This is the classic "normal distribution" (like heights of people or test scores). They found that for these shapes, the relationship between total and worst cost is very stable and predictable, regardless of how "wide" the bell curve is. It depends only on the dimension (how many directions the sand can move in).
Discrete Points: Imagine the sand isn't a continuous pile, but just a few distinct marbles. They proved that if you have a few marbles, you can still predict the worst-case distance based on the total cost, provided no single marble is too heavy (too much mass).

Why This Matters (According to the Paper)

The paper doesn't claim this will cure diseases or build better bridges immediately. Instead, it claims to provide a unified framework.

Before this, mathematicians had different rules for different situations (e.g., one rule for smooth sand, another for clumpy sand, another for 2D, another for 3D). This paper says: "We have one master formula that covers all these cases."

It acts like a universal translator for these types of mathematical problems. It tells us exactly how the "average" behavior of a system controls its "extreme" behavior, provided we know how the system is distributed.

Summary in a Nutshell

The Problem: Can we predict the worst-case distance between particles if we know the total energy?
The Context: Particles that repel each other (like magnets).
The Solution: Yes, but the prediction depends on how "clumped" the particles are.
The Result: A new mathematical inequality that connects the total cost to the maximum cost, valid for a wide variety of distributions (from smooth clouds to discrete dots).

The paper is essentially a rigorous proof that you can't hide a huge worst-case distance inside a small total cost if the particles are spread out nicely. If the total cost is low, the particles must be far apart from each other.

Technical Summary: Some reverse inequality in optimal mass transportation

Problem Statement
The paper addresses the problem of controlling the supremal (or $L^\infty$ ) optimal transport cost, denoted $C_\infty(\rho)$ , by the standard integral (or $L^1$ ) optimal transport cost, denoted $C(\rho)$ . Specifically, the authors investigate "reverse inequalities" of the form:
$\omega(C_\infty(\rho)) \leq C(\rho)$
where $\omega$ is an explicit, positive function for $t > 0$ .

While previous literature (notably [3] and [14]) has established such inequalities for attractive costs (where the cost function increases with distance, e.g., $c(x,y) = d(x,y)^p$ ), this paper focuses on repulsive costs. The cost function is defined as a sum of pairwise interactions $h(d(x_i, x_j))$ , where $h: [0, \infty) \to (0, \infty]$ is a continuous, non-increasing function. A primary model case considered is the Coulomb interaction where $h(t) = 1/t$ . The problem is particularly relevant when the marginals are identical ( $\mu = \nu = \rho$ ), a scenario where repulsive costs naturally arise.

Methodology
The authors work within the framework of the Kantorovich relaxation of the optimal transport problem on a complete, separable metric space (Polish space) $X$ . They consider $N$ -marginal transport plans $\lambda \in \Pi(\rho)$ , where $\Pi(\rho)$ is the set of probability measures on $X^N$ with all marginals equal to $\rho$ .

The core methodology involves:

Concentration Analysis: Introducing the concept of pointwise concentration $\kappa_\rho(\alpha) = \sup_{x \in X} \rho(B(x, \alpha))$ and analyzing the measure of sets where points are close to each other.
Set Estimates: Defining specific subsets of $X^N$ based on the cost threshold ( $B_B$ ) and the minimum distance between points ( $B_\beta$ ). The authors establish relationships between these sets, noting that for decreasing $h$ , small distances imply high costs.
Measure Lower Bounds: Proving that if the concentration of the measure $\rho$ exceeds a critical threshold ( $1/N$ ), then any transport plan $\lambda$ must assign positive mass to sets where points are close. Specifically, they derive lower bounds for $\lambda(B_{2\alpha})$ in terms of $\kappa_\rho(\alpha)$ .
Integration: Combining the lower bound on the measure of "close" configurations with the monotonicity of the cost function $h$ to bound the integral cost $C(\rho)$ from below using the supremal cost $C_\infty(\rho)$ .

Key Contributions and Results

General Reverse Inequality (Theorem 2.3): The main result provides a general lower bound for the integral cost $C(\rho)$ in terms of the supremal cost $C_\infty(\rho)$ for $N$ marginals. The inequality is:
$C(\rho) \geq h\left(2h^{-1}\left(\frac{C_\infty(\rho)}{N(N-1)}\right)\right) \left[ N\kappa_\rho\left(h^{-1}\left(\frac{C_\infty(\rho)}{N(N-1)}\right)\right) - \frac{1}{N(N-1)} \right]$
This result holds under the assumption that if $h(0+) = +\infty$ , then the pointwise concentration $\kappa(\rho) < 1/N$ .
Refinement for Two Marginals ( $N=2$ ): In Section 3, the authors restrict the analysis to $N=2$ . Using Fréchet bounds, they improve the lower bound on the measure of close configurations, obtaining a sharper inequality:
$C(\rho) \geq h\left(2h^{-1}\left(\frac{C_\infty(\rho)}{2}\right)\right) \left[ 2\kappa_\rho\left(h^{-1}\left(\frac{C_\infty(\rho)}{2}\right)\right) - 1 \right]$
Positivity of the Concentration Term: A critical technical step is ensuring the term in the brackets is strictly positive. Corollary 2.16 (and Corollary 3.5 for $N=2$ ) proves that the argument of the concentration function, $h^{-1}(\frac{C_\infty(\rho)}{N(N-1)})$ , is always large enough such that $\kappa_\rho(\cdot) > 1/N$ . This guarantees the validity of the inequality.
Classes of Measures with Uniform Constants: The paper acknowledges that a universal constant $C$ such that $C(\rho) \geq C \cdot C_\infty(\rho)$ does not hold for all measures (as shown in Example 1.3). However, Section 4 identifies specific classes of measures where such a constant exists, depending only on the class properties:
- Unimodal Isotropic Measures: For radially symmetric, unimodal, absolutely continuous measures (e.g., Gaussian, Cauchy, Laplace), the authors characterize $C_\infty(\rho)$ via the inverse of the diameter of a ball containing half the mass. They show that for these measures, the concentration term is bounded away from zero, yielding a constant $C_\rho$ dependent on the dimension but independent of the specific variance or scale.
- Discrete Measures: For discrete measures where the mass of any single atom is bounded below by $\delta > 0$ (and strictly less than $1/2$ ), the authors prove $C(\rho) \geq \frac{\delta}{2} C_\infty(\rho)$ .
Comparison with Existing Literature: The authors compare their derived concentration function $m(t)$ with the constant $\alpha$ derived from "trim" plans in discrete optimal transport (referencing [1]). They demonstrate that their general framework encompasses these discrete cases, providing a bound that is at least as strong as the one derived from the decomposition of trim minimizers.

Significance and Claims
The paper claims to establish a general framework that unifies the study of reverse inequalities for optimal transport with repulsive costs. By moving from the specific case of Wasserstein distances (attractive costs) to general decreasing cost functions, the authors extend the theoretical understanding of how integral and supremal transport costs relate.

The significance lies in:

Generalization: Extending the reverse inequality methodology to repulsive costs, which are crucial in models like Coulomb interactions.
Explicit Dependence: Providing an explicit formula for the function $\omega$ that relates $C$ and $C_\infty$ , which depends on the concentration properties of the measure $\rho$ .
Unification: Showing that the concentration-based approach naturally covers specific cases previously studied in the literature (such as discrete measures and specific attractive cost scenarios) while offering new insights for repulsive interactions.

The authors remain modest regarding the existence of a universal constant independent of the measure class, explicitly noting that such a constant cannot exist for arbitrary measures, but demonstrating that it is achievable for significant classes like unimodal isotropic distributions and discrete measures with bounded atomic masses.