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

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.