Quantitative rigidity of the Wasserstein contraction under convolution

The aim of this paper is to investigate the contraction properties of $p$-Wasserstein distances with respect to convolution in Euclidean spaces both qualitatively and quantitatively. We connect this question to the question of uniform convexity of the Kantorovich functional on which there was substantial recent progress (mostly for $p=2$ and partially for $p>1$). Motivated by this connection we extend these uniform convexity results to the case $p=1$, which is of independent interest.