  We tackle the problem of high-dimensional nonparametric density estimation by
taking the class of log-concave densities on $\mathbb{R}^p$ and incorporating
within it symmetry assumptions, which facilitate scalable estimation algorithms
and can mitigate the curse of dimensionality. Our main symmetry assumption is
that the super-level sets of the density are $K$-homothetic (i.e. scalar
multiples of a convex body $K \subseteq \mathbb{R}^p$). When $K$ is known, we
prove that the $K$-homothetic log-concave maximum likelihood estimator based on
$n$ independent observations from such a density has a worst-case risk bound
with respect to, e.g., squared Hellinger loss, of $O(n^{-4/5})$, independent of
$p$. Moreover, we show that the estimator is adaptive in the sense that if the
data generating density admits a special form, then a nearly parametric rate
may be attained. We also provide worst-case and adaptive risk bounds in cases
where $K$ is only known up to a positive definite transformation, and where it
is completely unknown and must be estimated nonparametrically. Our estimation
algorithms are fast even when $n$ and $p$ are on the order of hundreds of
thousands, and we illustrate the strong finite-sample performance of our
methods on simulated data.
