Let us consider convex polytopes with $K$ extreme points in $\mathbb{R}^d$. Let $\mathbf{P}$ be such polytope, and let $\text{ext}(\mathbf{P})$ denote its set of extreme points, so that $\mathbf{P} = \text{conv}(\text{ext}(\mathbf{P}))$ and $|\text{ext}(\mathbf{P})| =K$.
Let $\mathcal{L}|_{\mathbf{P}}$ be the uniform distribution on $\mathbf{P}$, or in other words, the restriction of $d$-dimensional Lebesgue measure to $\mathbf{P}$. Also, let $$\mathbb{P}_{\mathbf{P}} := \frac{1}{K} \sum_{x \in \text{ext}(\mathbf{P})} \delta_x$$ be the empirical measure on the extreme points of $\mathbf{P}$, that is the discrete uniform distribution on $\text{ext}(\mathbf{P})$.
Consider Wasserstein-$p$ distance between probability measures, for $p=1$ or $2$. We are interested in $$ W_p(\mathbb{P}_{\mathbf{P}},\mathcal{L}|_{\mathbf{P}}). $$ Is there a way to characterize it? For what polytopes $\mathbf{P}$, this distance is the smallest (largest)?
