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Synthetic MTW conditions and their equivalence under mild regularity assumption on the cost function

Seonghyeon Jeong

TL;DR

Addresses the equivalence of synthetic MTW conditions under $C^2$ regularity of the cost in optimal transport. The authors define Loeper's condition and QQconv, prove Loeper implies QQconv away from and near the boundary under structural assumptions, and establish the main theorem that Loeper's condition and QQconv are equivalent with only $C^2$ regularity. The work also highlights the necessity of the non-degeneracy assumption by furnishing a counterexample when it fails. This extends the MTW-equivalence from $C^4$ to $C^2$ costs, enabling Hölder regularity results for Monge-Ampère type equations in broader settings.

Abstract

Loeper's condition in \cite{Loe09} and the quantitatively quasi-convex condition (QQconv) from \cite{GK15} are synthetic expressions of the analytic MTW condition from \cite{TW} since they only require $C^2$ differentiability of the cost function $c$. When the cost function $c$ is $C^4$, it is known that the two synthetic MTW conditions are equivalent to the analytic MTW condition. However, when the cost function has regularity weaker than $C^4$, it is not known that if the two synthetic MTW conditions are equivalent. In this paper, we show the equivalence of the synthetic MTW conditions when the cost function has only $C^2$ regularity.

Synthetic MTW conditions and their equivalence under mild regularity assumption on the cost function

TL;DR

Addresses the equivalence of synthetic MTW conditions under regularity of the cost in optimal transport. The authors define Loeper's condition and QQconv, prove Loeper implies QQconv away from and near the boundary under structural assumptions, and establish the main theorem that Loeper's condition and QQconv are equivalent with only regularity. The work also highlights the necessity of the non-degeneracy assumption by furnishing a counterexample when it fails. This extends the MTW-equivalence from to costs, enabling Hölder regularity results for Monge-Ampère type equations in broader settings.

Abstract

Loeper's condition in \cite{Loe09} and the quantitatively quasi-convex condition (QQconv) from \cite{GK15} are synthetic expressions of the analytic MTW condition from \cite{TW} since they only require differentiability of the cost function . When the cost function is , it is known that the two synthetic MTW conditions are equivalent to the analytic MTW condition. However, when the cost function has regularity weaker than , it is not known that if the two synthetic MTW conditions are equivalent. In this paper, we show the equivalence of the synthetic MTW conditions when the cost function has only regularity.

Paper Structure

This paper contains 6 sections, 8 theorems, 104 equations.

Table of Contents

  1. introduction
  2. Assumptions on the cost function and domains
  3. The synthetic MTW conditions
  4. Interior QQconv
  5. Near the boundary
  6. A remark on a counter example

Key Result

Theorem 1.1

Let $X$ and $Y$ be compact subsets of $\mathbb{R}^n$, and let $c: X \times Y \to \mathbb{R}$ be a measurable function called the cost function. Assume that $c$ is twice differentiable in the sense that there exist the mixed Hessian $D^2_{xy} c$ and $D^2_{yx} c$, and the mixed Hessians are transpose

Theorems & Definitions (29)

  • Theorem 1.1: Main Theorem
  • Definition 2.1
  • Definition 2.1
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Remark 3.4
  • Remark 3.5
  • Remark 3.6
  • Lemma 3.7
  • ...and 19 more