Our method does converge since we follow coordinate descent and every step is convex [@grippo2000convergence], given the assumption we made in \[sec:ot\] that $\X, \Y \subset \mathbb{R}^{n}$, $c(\x, \y)= \|\x - \y\|_{2}$, and $p = 2$. There are in total $\mathcal{O}(\K \cdot \N)$ variables for computing $N$ Monge maps $\{\TL{i}\}_{i=1}^{\N}$, and $\mathcal{O}(\K)$ variables as the support $\bm{\y}$ and $\mathcal{O}(\K)$ variables as the measure $\bm{\NU}$. We implemented VWB with PyTorch [@paszke2019pytorch]. The code to reproduce the figures in this paper is at <https://github.com/icemiliang/pyvot>.

Metric Properties of (V)WBs {#sec:metric_property}
---------------------------

In spite of extensive studies on metric properties of OT over the past century, the metric properties of Wasserstein barycenters have yet been fully explored. Some pioneer work includes [@papadakis2019approximation; @auricchio2018computing].

However, most of them focus on the barycenter of two measures ($\N=2$). We show in the following that WBs in general ($\N \geq 2$) induce a *generalized metric* (*n-metric*). First, let us define the total Wasserstein distance between the barycenter and all the marginal Borel measures: $$\label{eq:wb_nmetric}
    \WBL{\NU}(\MUNL{1:N}) \eqdef \underset{\NU \in \MUS(\Y)}{\inf} \frac{1}{N}\sum_{i = 1}^{N} \WUL{2}{2}(\MUL{i}, \NU),$$ Then, we raise the following two propositions and prove them in Appendix.

\[the:wb\_nmetric\] $\WBL{\NU}(\MUNL{1:N})$ defines a generalized metric among $\{\MUL{i}\}_{i=1}^{\N}$, $N \geq 2$. Specifically, 