Sparsity and non-Euclidean embeddings

We present a relation between sparsity and non-Euclidean isomorphic embeddings. We introduce a general restricted isomorphism property and show how it enables to construct embeddings of $\ell_p^n$, $p > 0$, into various type of Banach or quasi-Banach spaces. In particular, for $0 <r < p<2$ with $r \le 1$, we construct a family of operators that embed $\ell_p^n$ into $\ell_r^{(1+\eta)n}$, with optimal polynomial bounds in $\eta >0$.