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

TL;DR

This paper proves the equivalence of two synthetic MTW conditions under mild regularity assumptions on the cost function, specifically when it is only twice differentiable, extending previous results that required higher regularity.

Contribution

The paper establishes the equivalence of Loeper's condition and QQconv for cost functions with only $C^2$ regularity, broadening the understanding of synthetic MTW conditions.

Findings

Equivalence of synthetic MTW conditions under $C^2$ regularity

Extension of previous equivalence results to lower regularity

Clarification of conditions needed for MTW equivalence

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.