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DCDS

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.

DCDS

To every distance $d$ on a given open set $\Omega\subseteq\mathbb R^n$,
we may associate several kinds of variational problems. We show
that, on the class of all geodesic distances $d$ on $\Omega$ which
are bounded from above and from below by fixed multiples of the
Euclidean one, the uniform convergence on compact sets turns out
to be equivalent to the $\Gamma$-convergence of each of the
corresponding variational problems under consideration.

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