  We study the adaptation properties of the multivariate log-concave maximum
likelihood estimator over three subclasses of log-concave densities. The first
consists of densities with polyhedral support whose logarithms are piecewise
affine. The complexity of such densities~$f$ can be measured in terms of the
sum $\Gamma(f)$ of the numbers of facets of the subdomains in the polyhedral
subdivision of the support induced by $f$. Given $n$ independent observations
from a $d$-dimensional log-concave density with $d \in \{2,3\}$, we prove a
sharp oracle inequality, which in particular implies that the Kullback--Leibler
risk of the log-concave maximum likelihood estimator for such densities is
bounded above by $\Gamma(f)/n$, up to a polylogarithmic factor. Thus, the rate
can be essentially parametric, even in this multivariate setting. For the
second type of adaptation, we consider densities that are bounded away from
zero on a polytopal support; we show that up to polylogarithmic factors, the
log-concave maximum likelihood estimator attains the rate $n^{-4/7}$ when
$d=3$, which is faster than the worst-case rate of $n^{-1/2}$. Finally, our
third type of subclass consists of densities whose contours are well-separated;
these new classes are constructed to be affine invariant and turn out to
contain a wide variety of densities, including those that satisfy H\"older
regularity conditions. Here, we prove another sharp oracle inequality, which
reveals in particular that the log-concave maximum likelihood estimator attains
a risk bound of order
$n^{-\min\bigl(\frac{\beta+3}{\beta+7},\frac{4}{7}\bigr)}$ when $d=3$ over the
class of $\beta$-H\"older log-concave densities with $\beta\in (1,3]$, again up
to a polylogarithmic factor.
