/ \cos r_{\IdxCentroid}$ and thus the power Voronoi diagram in the spherical domain $\RegionL{\IdxCentroid} \eqdef \{\x \in \mathbb{S}^{2}\ | c'(\x, \yL{\IdxCentroid}) \leq c'(\x, \yL{\IdxCentroidSecond}), \forall \IdxCentroidSecond \neq \IdxCentroid\}$. $r_{\IdxCentroid}$ is the weight of each power cell, it relates to the VOT minimizers by $\cos{r} = e^{h}$. Then, the derivation in \[sec:vot\] gives us the Monge map.

$-\ln\langle\x , \yL{\IdxCentroid}\rangle$ does not satisfy triangle inequality but the other three metric properties. Thus, $\WL{1}'$ inherits those properties. We notice that the proof for Proposition \[the:wb\_nmetric\] does not require triangle inequality. Therefore, the n-metric properties still hold for the barycenter w.r.t. $\WL{1}'$. $$\WBL{1:\N}'(\NU) \eqdef \underset{\NU \in \MUS(\Y)}{\inf} \frac{1}{\N}\sum_{i = 1}^{\N} \WL{1}'(\MUL{i}, \NU)$$ Although $\WL{1}'$ is not a true metric, we can still find a “mean” of multiple marginals by alternatively minimizing the total “distance” as in \[sec:wbvot\]. Figure \[fig:sphere\] shows an example where the VWB simultaneously partitions two domains on the sphere. For simplicity, we draw connections with straight lines.

Geometric Clustering via VWBs {#sec:gcvwb}
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In this section, we further connect VWBs to several clustering problems. We consider a fixed number of clusters, $K$, the quadratic Euclidean distance as the ground metric, and mainly the spatial relation between samples. We refer to 