Embeddability of snowflaked metrics with applications to the nonlinear geometry of the spaces L_p and ell_(p) for 0<p<infty

We study the classical spaces $L_{p}$ and $\ell_{p}$ for the whole range $0<p<\infty$ from a metric viewpoint and give a complete Lipschitz embeddability roadmap between any two of those spaces when equipped with both their ad-hoc distances and their snowflakings. Through connections with weaker forms of embeddings that lead to basic (yet fundamental) open problems, we also set the challenging goal of understanding the dissimilarities between the well-known subspace structure and the different nonlinear geometries that coexist inside $L_{p}$ and $\ell_{p}$.