respect to the inner product $\langle \cdot , \cdot \rangle_{\rho_k}$. This may as well be defined as the first non-trivial eigenvalue of the operator $\Delta_{\rho_k}$.

\[def:well-separated-mixture\] A mixture model $$\rho =  \sum_{k=1}^N w_k \rho_k$$ is well-separated if $$\mathcal{S} \ll 1 , \quad \frac{{\mathcal{C}}}{{\Theta}} \ll 1.$$ These conditions are to be read as: the overlapping ${\mathcal{S}}$ is small enough, and the coupling parameter $\mathcal{C}$ is small in comparison to the indivisibility parameter ${\Theta}$.

In Section \[sec:discussionparameters\] we discuss different interpretations of ${\mathcal{S}}, {\mathcal{C}}, {\Theta}$ and present some examples whose purpose is to illustrate the significance of all three parameters. Notice that the notion of a well-separated model has been introduced qualitatively, matching the informal version of our first main result (Theorem \[thm:well-separated-gives-cone-structure-continuum\]). In Theorem \[mainTheorem\] we give a precise quantitative relation of how tightly the measure $F_{\sharp} \nu$ concentrates around orthogonal vectors depending on the quantities ${\mathcal{S}}$ and ${\mathcal{C}}/\Theta$. In other words, Theorem \[mainTheorem\] is the quantitative (and precise) version of Theorem \[thm:well-separated-gives-cone-structure-continuum\].

We will use the functions $q_1, \dots, q_N$ frequently in the remainder so it is worth mentioning a simple interpretation for them. Suppose that we wanted to obtain one sample from the mixture model introduced in : ${\mathbf{x}}\sim 