instead. We introduce an energy: $$\label{eq:vot}
        \ignore{\min_{\bm{\h}}}\ \EnergyL{2}[\bm{\h}] \eqdef \int_{\bm{0}}^{\bm{\h}} \sum_{\IdxCentroid=1}^{\K} \int_{\RegionL{\IdxCentroid}} d\MU(\x) d\hL{\IdxCentroid} - \sum_{\IdxCentroid = 1}^{\K}\NU(\yL{\IdxCentroid})\hL{\IdxCentroid},$$ whose gradient, $\big\{\int_{\RegionL{\IdxCentroid}} d\MU(\x) - \NU(\yL{\IdxCentroid})\big\}_{\IdxCentroid}$, also integrates to $$\label{eq:theta}
    \ignore{\min_{\bm{\h}}}\ \EnergyL{3}[\bm{\h}] \eqdef \int_{\X} \theta_{\bm{\h}}(\x) d\MU(\x) - \sum_{\IdxCentroid = 1}^{\K}\NU(\yL{\IdxCentroid})\hL{\IdxCentroid}.$$ Meanwhile, the *Lagrangian duality* of Monge OT is $$\label{eq:monge_dual}
\begin{gathered}
    \max_{\bm{\LagPhi}}\ \min_{\T}\ \EnergyL{4}[\bm{\LagPhi},T] 
    \eqdef \\ \int_{\X} \big( \|\x - \T(\x)\|_{2}^{2} + \sum_{\IdxCentroid=1}^{\K} \LagPhiL{\IdxCentroid} \big) d\MU(x)  - \sum_{\IdxCentroid = 1}^{\K} \LagPhiL{\IdxCentroid}\NU(\yL{\IdxCentroid}),
\end{gathered}$$ where $\bm{\LagPhi} = \{\varphi_\IdxCentroid\}_{\IdxCentroid=1}^{\K}$. simplifies to $$\label{eq:monge_ot_dual}
\begin{gathered}
    \max_{\bm{\LagPhi}}\ \EnergyL{4}[\bm{\LagPhi}] \\
    = \sum_{\IdxCentroid = 1}^{\K} \int_{\RegionL{\IdxCentroid}'} \big(\|\x - \yL{\IdxCentroid}\|_{2}^{2} + \LagPhiL{\IdxCentroid} \big) d\MU(\x) - \sum_{\IdxCentroid = 1}^{\K} \LagPhiL{\IdxCentroid}\NU(\yL{\IdxCentroid}),
\end{gathered}$$ $\RegionL{\IdxCentroid}' = \{\x \in \X\ |\ \|\x - \yL{\IdxCentroid}\|_{2}^{2} + \LagPhiL{\IdxCentroid} \leq \|\x - \yL{\IdxCentroidSecond}\|_{2}^{2} + \LagPhiL{\IdxCentroidSecond}, \forall \IdxCentroidSecond \neq \IdxCentroid\}$ which coincides with a *power Voronoi diagram*.

We provide detailed derivation for above formulas in Appendix and then prove their following connections.

\[th:energy\_connection\] **1**. The minimum point of $\EnergyL{2}[\bm{\h}]$, , also minimizes $\EnergyL{3}[\bm{\h}]$, . **2**. $\RegionL{\IdxCentroid} \equiv \RegionL{\IdxCentroid}'$. **3**. $\bm{\Region}$ in $\EnergyL{2}[\bm{\h}]$, , induces the Monge map $\T: x \rightarrow \yL{\IdxCentroid}$. **4**. Minimizing $\EnergyL{2}[\bm{\h}]$, , is equivalent to maximizing $\EnergyL{4}[\bm{\h}]$, .

Therefore, we “variationally” minimize $\EnergyL{2}[\bm{\h}]$, , for a *height vector* $\bm{\h}$ and that will produce a Monge map $\TU{*}$.

Wasserstein 