  Pinsker's widely used inequality upper-bounds the total variation distance
$||P-Q||_1$ in terms of the Kullback-Leibler divergence $D(P||Q)$. Although in
general a bound in the reverse direction is impossible, in many applications
the quantity of interest is actually $D^*(P,\eps)$ --- defined, for an
arbitrary fixed $P$, as the infimum of $D(P||Q)$ over all distributions $Q$
that are $\eps$-far away from $P$ in total variation. We show that
$D^*(P,\eps)\le C\eps^2 + O(\eps^3)$, where $C=C(P)=1/2$ for "balanced"
distributions, thereby providing a kind of reverse Pinsker inequality. An
application to large deviations is given, and some of the structural results
may be of independent interest. Keywords: Pinsker inequality, Sanov's theorem,
large deviations
