  Given an isotropic random vector $X$ with log-concave density in Euclidean
space $\Real^n$, we study the concentration properties of $|X|$ on all scales,
both above and below its expectation. We show in particular that: \[
\P(\abs{|X| -\sqrt{n}} \geq t \sqrt{n}) \leq C \exp(-c n^{1/2} \min(t^3,t))
\;\;\; \forall t \geq 0 ~, \] for some universal constants $c,C>0$. This
improves the best known deviation results on the thin-shell and mesoscopic
scales due to Fleury and Klartag, respectively, and recovers the sharp
large-deviation estimate of Paouris. Another new feature of our estimate is
that it improves when $X$ is $\psi_\alpha$ ($\alpha \in (1,2]$), in precise
agreement with Paouris' estimates. The upper bound on the thin-shell width
$\sqrt{\Var(|X|)}$ we obtain is of the order of $n^{1/3}$, and improves down to
$n^{1/4}$ when $X$ is $\psi_2$. Our estimates thus continuously interpolate
between a new best known thin-shell estimate and the sharp large-deviation
estimate of Paouris. As a consequence, a new best known bound on the Cheeger
isoperimetric constant appearing in a conjecture of
Kannan--Lov\'asz--Simonovits is deduced.
