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On the equivalence of $c$-potentiability and $c$-path boundedness in the sense of Artstein-Avidan, Sadovsky, and Wyczesany

Sedi Bartz, Heinz H. Bauschke, Yuan Gao

TL;DR

The paper extends the classical Rockafellar framework to costs that may take the value $+\infty$, by analyzing when $c$-path boundedness implies $c$-potentiability. It develops a general potentiability theorem free of topology, and shows sufficiency results in separable metric spaces under continuity, supplemented by the novel notion of a $c$-path bounded extension to handle complex connectivity (including infinite black holes). The approach unifies several nontraditional costs (polar, Coulomb, Bregman) under a common mechanism, and provides constructive avenues to obtain $c$-potentials via $c$-antiderivatives and chain extensions. These results deepen the understanding of when $c$-cyclic monotonicity leads to potentiability and thus to Kantorovich duality in broader cost landscapes, with potential impact on optimal transport in physics, geometry, and economics.

Abstract

A cornerstone of convex analysis, established by Rockafellar in 1966, asserts that a set has a potential if and only if it is cyclically monotone. This characterization was generalized to hold for any real-valued cost function $c$ and lies at the core structure of optimal transport plans. However, this equivalence fails to hold for costs that attain infinite values. In this paper, we explore potentiability for an infinite-valued cost $c$ under the assumption of $c$-path boundedness, a condition that was first introduced by Artstein-Avidan, Sadovsky and Wyczesany. This condition is necessary for potentiability and is more restrictive than $c$-cyclic monotonicity. We provide general settings and other conditions under which $c$-path boundedness is sufficient for potentability, and therefore equivalent. We provide a general theorem for potentiability, requiring no topological assumptions on the spaces or the cost. We then provide sufficiency in separable metric spaces and costs that are continuous in their domain. Finally, we introduce the notion of a $c$-path bounded extension and use it to prove the existence of potentials for a special class of costs on $\mathbb{R}^2$. We illustrate our discussion and results with several examples.

On the equivalence of $c$-potentiability and $c$-path boundedness in the sense of Artstein-Avidan, Sadovsky, and Wyczesany

TL;DR

The paper extends the classical Rockafellar framework to costs that may take the value , by analyzing when -path boundedness implies -potentiability. It develops a general potentiability theorem free of topology, and shows sufficiency results in separable metric spaces under continuity, supplemented by the novel notion of a -path bounded extension to handle complex connectivity (including infinite black holes). The approach unifies several nontraditional costs (polar, Coulomb, Bregman) under a common mechanism, and provides constructive avenues to obtain -potentials via -antiderivatives and chain extensions. These results deepen the understanding of when -cyclic monotonicity leads to potentiability and thus to Kantorovich duality in broader cost landscapes, with potential impact on optimal transport in physics, geometry, and economics.

Abstract

A cornerstone of convex analysis, established by Rockafellar in 1966, asserts that a set has a potential if and only if it is cyclically monotone. This characterization was generalized to hold for any real-valued cost function and lies at the core structure of optimal transport plans. However, this equivalence fails to hold for costs that attain infinite values. In this paper, we explore potentiability for an infinite-valued cost under the assumption of -path boundedness, a condition that was first introduced by Artstein-Avidan, Sadovsky and Wyczesany. This condition is necessary for potentiability and is more restrictive than -cyclic monotonicity. We provide general settings and other conditions under which -path boundedness is sufficient for potentability, and therefore equivalent. We provide a general theorem for potentiability, requiring no topological assumptions on the spaces or the cost. We then provide sufficiency in separable metric spaces and costs that are continuous in their domain. Finally, we introduce the notion of a -path bounded extension and use it to prove the existence of potentials for a special class of costs on . We illustrate our discussion and results with several examples.

Paper Structure

This paper contains 11 sections, 29 theorems, 130 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Auxiliary results
  3. $c$-antiderivatives vs $c$-potentials
  4. Paths and maximal inner variations
  5. $c$-antiderivative of costs with disconnected domain
  6. A simple direct proof of \ref{['f:fix']}.(i)
  7. $c$-antiderivatives in the general setting
  8. $c$-antiderivatives in the metric setting with open domain
  9. Monotone costs and $c$-path bounded extensions in $\mathbb{R}^2$
  10. $c$-path bounded extension for $\ominus$-monotone costs
  11. $c$-path bounded extension for $\oplus$-monotone costs

Key Result

Corollary 1.15

Let $G$ be a nonempty strongly connected subset of $D$. Then $G$ is $c$-potentiable $\Leftrightarrow$$G$ is $c$-path bounded.

Figures (1)

  • Figure 1: The green area is the domain of $c$ in the first quadrant. $G$ consists of the blue segments.

Theorems & Definitions (112)

  • Definition 1.1: $c$-conjugates and $c$-class functions
  • Definition 1.2: $c$-subdifferential
  • Remark 1.3
  • Remark 1.4: convex analysis setting
  • Definition 1.5: path, condensation graph, and strongly connected components
  • Definition 1.6: $c$-path boundedness and $c$-cyclic monotonicity
  • Definition 1.7: $c$-potentiability
  • proof
  • proof
  • Remark 1.10: Rockafellar-Rochet-Rüschendorf
  • ...and 102 more