#### Vol. 12, No. 1, 2021

 Recent Issues Volume 12, Issue 1 Volume 11, Issue 2 Volume 11, Issue 1
 The Journal About the Journal Editorial Board Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN (electronic): 2693-3004 ISSN (print): 2693-2997 To Appear Other MSP Journals
1-Wasserstein distance on the standard simplex

### Andrew Frohmader and Hans Volkmer

Vol. 12 (2021), No. 1, 43–56
##### Abstract

Wasserstein distances provide a metric on a space of probability measures. We consider the space $\Omega$ of all probability measures on the finite set $\chi =\left\{1,\dots ,n\right\}$, where $n$ is a positive integer. The 1-Wasserstein distance, ${W}_{1}\left(\mu ,\nu \right)$, is a function from $\Omega ×\Omega$ to $\left[0,\infty \right)$. This paper derives closed-form expressions for the first and second moments of ${W}_{1}$ on $\Omega ×\Omega$ assuming a uniform distribution on $\Omega ×\Omega$.

##### Keywords
Wasserstein, earth mover, probability simplex, simplex, standard simplex, moments, expected value, variance
##### Mathematical Subject Classification 2010
Primary: 28A33, 60B05