  $ \renewcommand{\subset}{\subseteq} \newcommand{\N}{\mathbb N} $For $p\in
[2,\infty)$ the metric $X_p$ inequality with sharp scaling parameter is proven
here to hold true in $L_p$. The geometric consequences of this result include
the following sharp statements about embeddings of $L_q$ into $L_p$ when $2<
q<p<\infty$: the maximal $\theta\in (0,1]$ for which $L_q$ admits a
bi-$\theta$-H\"older embedding into $L_p$ equals $q/p$, and for $m,n\in \N$ the
smallest possible bi-Lipschitz distortion of any embedding into $L_p$ of the
grid $\{1,\ldots,m\}^n\subset \ell_q^n$ is bounded above and below by constant
multiples (depending only on $p,q$) of the quantity
$\min\{n^{(p-q)(q-2)/(q^2(p-2))}, m^{(q-2)/q}\}$.
