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Loeper's condition in \cite{Loe09} and the quantitatively quasi-convexcondition (QQconv) from \cite{GK15} are synthetic expressions of the analyticMTW condition from \cite{TW} since they only require $C^2$ differentiability ofthe cost function $c$. When the cost function $c$ is $C^4$, it is known thatthe two synthetic MTW conditions are equivalent to the analytic MTW condition.However, when the cost function has regularity weaker than $C^4$, it is notknown that if the two synthetic MTW conditions are equivalent. In this paper,we show the equivalence of the synthetic MTW conditions when the cost functionhas regularity slightly weaker than $C^3$.

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