  In this paper, we study two problems: (1) estimation of a $d$-dimensional
log-concave distribution and (2) bounded multivariate convex regression with
random design with an underlying log-concave density or a compactly supported
distribution with a continuous density.
  First, we show that for all $d \ge 4$ the maximum likelihood estimators of
both problems achieve an optimal risk of $\Theta_d(n^{-2/(d+1)})$ (up to a
logarithmic factor) in terms of squared Hellinger distance and $L_2$ squared
distance, respectively. Previously, the optimality of both these estimators was
known only for $d\le 3$. We also prove that the $\epsilon$-entropy numbers of
the two aforementioned families are equal up to logarithmic factors. We
complement these results by proving a sharp bound $\Theta_d(n^{-2/(d+4)})$ on
the minimax rate (up to logarithmic factors) with respect to the total
variation distance.
  Finally, we prove that estimating a log-concave density - even a uniform
distribution on a convex set - up to a fixed accuracy requires the number of
samples \emph{at least} exponential in the dimension. We do that by improving
the dimensional constant in the best known lower bound for the minimax rate
from $2^{-d}\cdot n^{-2/(d+1)}$ to $c\cdot n^{-2/(d+1)}$ (when $d\geq 2$).
