q_k := \sqrt{\frac{w_k \rho_k}{\rho}}.$$ The *overlapping* parameter of the mixture model $\nu$ is the quantity $${\mathcal{S}}:=   \max_{i \not = j}  \left \langle \left(\frac{q_i}{\sqrt{w_i}}\right)^2 , \left(\frac{q_j}{\sqrt{w_j}} \right)^2\right\rangle_\rho    =  \max_{i \not = j}   \int_{{\mathcal{M}}} \frac{\rho_i \rho_j}{\rho} dx,
\label{eqn:S2}$$ Note that ${\mathcal{S}}\in\left(0,\frac{1}{\min_j w_j}\right)$ since for any $k\in\{1,...,N\}$ we have the bound $$(\min_j w_j) \rho_k\leq \sum_{j=1}^N w_j \rho_j =\rho\,,$$ and so $\frac{\rho_k}{\rho}\leq \frac{1}{\min_jw_j}$.

The *coupling* parameter of the mixture model is defined as $$\mathcal{C}:= \max_{k=1, \dots, N} \mathcal{C}_k,
\label{Coupling-parameter}$$ where $$\mathcal{C}_k:= 
\frac{1}{4} \int_{{\mathcal{M}}}  \left|\frac{\nabla\rho_k}{\rho_k}-\frac{\nabla\rho}{\rho}\right|^2\rho_k dx ,
\quad k=1, \dots, N.
 \label{Coupling}$$ This quantity is the Fisher information of the measure $\rho_kdx$ with respect to the measure $\rho dx$. We use the convention that if the density $\rho_k(x)$ vanishes at some point $x \in {\mathcal{M}}$, then we interpret the integrand in as zero at that point; in other words we only integrate over the set $\{ x \in {\mathcal{M}}\: : \: \rho_k(x) >0 \}$.

Finally, the *indivisibility* parameter for the model is defined as $$\label{Lambda-definition}
{\Theta}:= \min_{k=1, \dots, N} {\Theta}_k,$$ where $${\Theta}_k :=  \min_{u \perp \mathds{1}} \frac{\int_{{\mathcal{M}}} \lvert \nabla u \rvert^2 \rho_k dx }{\langle  u,  u\rangle_{\rho_k}}.
\label{Indivisbility}$$ In , $\mathds{1}$ stands for the function that is identically equal to one, and $\perp$ indicates orthogonality with 