Let $n,N\in\mathbb N$ with $N\ge 2$. Let $\Omega\subset\mathbb R^n$ be compact and convex with nonempty interior. Let $\mu_0$ be the Borel probability measure on $\Omega$ defined by
for every $B\in\mathcal B(\Omega)$, where $\rho_0\in C(\Omega;[0,\infty))$ and there exist constants $0<c\le C<\infty$ such that
paragraph
admin
\begin{align*}
c\le \rho_0(x)\le C
\end{align*}
latex_env
admin
for every $x\in\Omega$. Let $y_1,\dots,y_N\in\mathbb R^n$ be distinct points. For every $z=(z_1,\dots,z_N)\in\mathbb R^N$ and every $i\in\{1,\dots,N\}$, define the Laguerre cell
paragraph
admin
\begin{align*}
P_i(z)=\{x\in\Omega: |x-y_i|^2-z_i\le |x-y_j|^2-z_j \text{ for every } j\in\{1,\dots,N\}\}.
\end{align*}
latex_env
admin
Define the semi-discrete mass map $\mathcal M:\mathbb R^N\to\mathbb R^N$ by
Assume that $G_w$ is connected. Assume moreover that there is an open neighbourhood $W\subset\mathbb R^N$ of $w$ such that $\mathcal M|_W:W\to\mathbb R^N$ is $C^1$ and the Jacobian matrix $J\mathcal M_w$ is the signed weighted graph Laplacian of $G_w$: there exist symmetric weights $a_{ij}=a_{ji}\ge 0$ such that $a_{ij}>0$ exactly on the edges of $G_w$, $a_{ii}=0$, and
\begin{align*}
\Delta_N^\circ=\{q\in A:q_i>0\text{ for every }i\in\{1,\dots,N\}\}.
\end{align*}
latex_env
admin
If $m=\mathcal M(w)$, then there exist neighbourhoods $U\subset H$ of $w$ and $V\subset A$ of $m$ such that
paragraph
admin
\begin{align*}
\mathcal M|_U:U\to V
\end{align*}
latex_env
admin
is bijective and has a continuous local inverse. Consequently, every mass vector $m'\in V\cap\Delta_N^\circ$ is realized by a unique normalized weight $w'\in U$ satisfying