Wasserstein Rigidity over ${\mathbb{R}}^n$ with smooth norms

We study $p-$Wasserstein spaces ${\mathcal{W}}_p({\mathbb{R}}^n,d_N)$ over ${\mathbb{R}}^n$ equipped with a norm metric $d_N$. We show that, if the norm is smooth enough, then the Wasserstein space is isometrically rigid whenever $p\mathrm{\ne }2$. We also show that, even when $p=2$, we can recover the isometric rigidity of the Wasserstein space ${\mathcal{W}}_2({\mathbb{R}}^n,d_N)$ when $N$ is an $l_q-$norm and $q>2$.