(which ultimately induce corresponding spectral embeddings). The model considered in that paper is more geometric in nature than the one we consider here. In [@LingStromer], the authors introduce a notion of well-separated components at the finite data level and use it to validate a new algorithm for clustering which is based on a convex relaxation for a multi-way cut problem and does not require a $k$-means step, making it quite appealing.

We believe it is worth exploring and understanding the connections of the many different mathematical ideas and models presented in [@LittleMaggioniMurphy; @LingStromer] with the ones we present here.

Outline
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The rest of the paper is divided as follows. Section \[Set-up\] introduces the precise set-up of the paper and presents the main results. In particular, in Section \[sec:wellsepa\] we introduce our notion of well-separated mixture models; our two main results, Theorem \[mainTheorem\] on the geometry of continuum Laplacian embeddings (i.e. the measure clustering embedding) and Theorem \[PropMuMun\] describing the geometry of graph Laplacian embeddings, are presented in Sections \[sec:continuum\] and \[sec:fromconttodiscr\] respectively.

Section \[sec:discussion\] is devoted to discussing our notion of well-separated mixture models. We present some illustrative examples, discuss the implications of our results, and finally comment on 