$\WBL{\NU}(\MUNL{1:N})$ satisfies the following properties.\
1) Non-negativity: $\WBL{\NU}(\MUNL{1:N}) \geq 0$.\
2) Symmetry: $\WBL{\sigma_1(1:\N)}(\NU) = \WBL{\sigma_2(1:\N)}(\NU)$, where $\sigma_1(1:N)$ and $\sigma_2(1:\N)$ are different permutations of the set ${1:\N}$.\
3) Identity: $\WBL{\NU}(\MUNL{1:N}) = 0 \Longleftrightarrow \MUL{i} = \MUL{j}, \forall i \neq j$.\
4) Triangle inequality: $\WBL{\NU}(\MUNL{1:N}) \leq \sum_{i=1}^{\N} \WBL{\NU}(\MUNL{1:\N+1 \backslash i})$.

\[the:wb\_nmetric\_bigon\] The bound of the triangle inequality in Proposition \[the:wb\_nmetric\] can be tightened by a linear factor. Specifically, we have $(N/2)\ \WBL{\NU}(\MUNL{1:N}) \leq \sum_{i=1}^{\N} \WBL{\NU}(\MUNL{1:\N+1 \backslash i})$.

The VWB $\NU = \sum_{\IdxCentroid=1}^{\K}\NUL{j}\delta[\yL{\IdxCentroid}] \in \MUS(\Y)$, as a special case of WBs, certainly inherits the metric properties because there is not such a restriction on the continuity of $\Y$. If we denote the total WD for the VWB with $\VWBL{\NU}(\MUNL{1:\N})$ , then we have:

\[the:vwb\_nmetric\]

$\VWBL{\NU}(\MUNL{1:\N})$ induces an n-metric over all $\MUL{i}$’s. In particular, the equal signs in 1) non-negativity and 4) inequality hold if and only if all $\MUL{i}$’s and $\NU$ have the same number of supports with positive Dirac measures $|\MUL{i}| = |\NU| = \K,\ \forall i \in \{1,...,\N\}$.

![Transshipment: transporting measures through a set of discrete relays. Colors on the measures specify correspondences.[]{data-label="fig:ship1"}](ship1.pdf){width="0.9\columnwidth"}

Approximate WDs with VWBs – Transshipment {#sec:transship}
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We consider the transshipment problem as finding a Monge map 