  In this paper fundamental nonlinear geometries of Lebesgue sequence spaces
are studied in their quantitative aspects. Applications of this work are a
positive solution to the strong embeddability problem from $\ell_q$ into
$\ell_p$ ($0<p<q\le 1$) and new insights on the coarse embedabbility problem
from $L_q$ into $\ell_q$, $q>2$. Relevant to geometric group theory purposes,
the exact $\ell_p$-compressions of $\ell_2$ are computed. Finally coarse
deformation of metric spaces with property A and locally compact amenable
groups is investigated.
