论文标题
合成的MTW条件及其在轻度规律性假设对成本函数下的等效性
Synthetic MTW conditions and their equivalence under mild regularity assumption on the cost function
论文作者
论文摘要
\ cite {loe09}中的条件和来自\ cite {gk15}的定量Quasi-convex条件（QQConv）是\ cite {tw}的分析MTW条件的合成表达式，因为它们仅需要成本函数$ C^2 $ C^2 $ c $ c $ C $ C $ C^2 $。当成本函数$ c $为$ c^4 $时，众所周知，两个合成的MTW条件相当于分析MTW条件。但是，当成本函数的规律性弱于$ c^4 $时，尚不清楚，如果两个合成的MTW条件等效。在本文中，当成本函数的规律性略低于$ c^3 $时，我们显示了合成MTW条件的等效性。
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 regularity slightly weaker than $C^3$.
