  The log-concave maximum likelihood estimator (MLE) problem answers: for a set
of points $X_1,...X_n \in \mathbb R^d$, which log-concave density maximizes
their likelihood? We present a characterization of the log-concave MLE that
leads to an algorithm with runtime $poly(n,d, \frac 1 \epsilon,r)$ to compute a
log-concave distribution whose log-likelihood is at most $\epsilon$ less than
that of the MLE, and $r$ is parameter of the problem that is bounded by the
$\ell_2$ norm of the vector of log-likelihoods the MLE evaluated at
$X_1,...,X_n$.
