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

Seonghyeon Jeong

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arXiv · 2010.14471

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

Seonghyeon Jeong

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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.

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Seonghyeon Jeong. 2020-10-27. Synthetic MTW conditions and their equivalence under mild regularity assumption on the cost function. https://arxiv.org/abs/2010.14471

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