possible extensions and generalizations.

Proofs of the main results are postponed to Section \[sec:Proofs\]. In particular, Section \[sec:Conic\] focuses on stability properties of orthogonal cone structures, allowing us to prove Theorem \[mainTheorem\] in Section \[sec:ProofsMain\] after showing several preliminary results. In Section \[App:A\] we make the connection between the eigenspaces generated by the first $N$ eigenvectors of the graph Laplacian and those of the continuum Laplacian. We conclude with Section \[sec:proof2\], where we present the proof of Theorem \[PropMuMun\].

Setup and main results {#Set-up}
======================

Let $\mathcal{M}$ be a smooth, connected, orientable, $m$-dimensional manifold embedded in ${\mathbb{R}}^d$. For the moment we think of ${\mathcal{M}}$ as either a compact manifold without boundary or the whole of ${\mathbb{R}}^d$. Let $\rho_1, \dots, \rho_N$ be a family of $C^1({\mathcal{M}})$ probability density functions on ${\mathcal{M}}$ (densities with respect to the volume form of ${\mathcal{M}}$) and let $w_1,\dots, w_N$ be positive weights adding to one. The densities $\rho_1, \dots, \rho_N$ are weighted to produce the *mixture model* $$\rho(x) := \sum_{k=1}^{N} w_k \rho_k(x) , \quad  x \in {\mathcal{M}}.
\label{DefRho}$$ We assume that $\rho(x)>0$ for all $x \in {\mathcal{M}}$ and let $\nu$ be the probability measure on ${\mathcal{M}}$ induced by $\rho$, that is, $$d \nu(x) = \rho(x) dx 