  We study the approximation of arbitrary distributions $P$ on $d$-dimensional
space by distributions with log-concave density. Approximation means minimizing
a Kullback--Leibler-type functional. We show that such an approximation exists
if and only if $P$ has finite first moments and is not supported by some
hyperplane. Furthermore we show that this approximation depends continuously on
$P$ with respect to Mallows distance $D_1(\cdot,\cdot)$. This result implies
consistency of the maximum likelihood estimator of a log-concave density under
fairly general conditions. It also allows us to prove existence and consistency
of estimators in regression models with a response $Y=\mu(X)+\epsilon$, where
$X$ and $\epsilon$ are independent, $\mu(\cdot)$ belongs to a certain class of
regression functions while $\epsilon$ is a random error with log-concave
density and mean zero.
