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

Optimality of Maximum Likelihood for Log-Concave Density Estimation and Bounded Convex Regression