On Sudakov'S Type Decomposition Of Transference Plans With Norm Costs

The authors consider the original strategy proposed by Sudakov for solving the Monge transportation problem with norm cost $|\cdot|_{D^*}$$\min \bigg\{ \int |\mathtt T(x) - x|_{D^*} d\mu(x), \ \mathtt T : \mathbb{R}^d \to \mathbb{R}^d, \ \nu = \mathtt T_\# \mu \bigg\},$ with $\mu$, $\nu$ probability measures in $\mathbb{R}^d$ and $\mu$ absolutely continuous w.r.t. $\mathcal{L}^d$.