On the twist condition and $c$-monotone transport plans
Thierry Champion Luigi De Pascale
A usual approach for proving the existence of an optimal transport map, be it in ${\mathbb R}^d$ or on more general manifolds, involves a regularity condition on the transport cost (the so-called Left Twist condition, i.e. the invertibility of the gradient in the first variable) as well as the fact that any optimal transport plan is supported on a cyclically-monotone set. Under the classical assumption that the initial measure does not give mass to sets with $\sigma$-finite $\mathcal{H}^{d-1}$ measure and a stronger regularity condition on the cost (the Strong Left Twist), we provide a short and self-contained proof of the fact that any feasible transport plan (optimal or not) satisfying a $c$-monotonicity assumption is induced by a transport map. We also show that the usual costs induced by Tonelli Lagrangians satisfy the Strong Left Twist condition we propose.
keywords: Tonelli Lagrangian. cyclical monotonicity optimal transport problem Monge-Kantorovich problem

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