  We find limiting distributions of the nonparametric maximum likelihood
estimator (MLE) of a log-concave density, that is, a density of the form
$f_0=\exp\varphi_0$ where $\varphi_0$ is a concave function on $\mathbb{R}$.
The pointwise limiting distributions depend on the second and third derivatives
at 0 of $H_k$, the "lower invelope" of an integrated Brownian motion process
minus a drift term depending on the number of vanishing derivatives of
$\varphi_0=\log f_0$ at the point of interest. We also establish the limiting
distribution of the resulting estimator of the mode $M(f_0)$ and establish a
new local asymptotic minimax lower bound which shows the optimality of our mode
estimator in terms of both rate of convergence and dependence of constants on
population values.
