  We present theoretical properties of the log-concave maximum likelihood
estimator of a density based on an independent and identically distributed
sample in $\mathbb{R}^d$. Our study covers both the case where the true
underlying density is log-concave, and where this model is misspecified. We
begin by showing that for a sequence of log-concave densities, convergence in
distribution implies much stronger types of convergence -- in particular, it
implies convergence in Hellinger distance and even in certain exponentially
weighted total variation norms. In our main result, we prove the existence and
uniqueness of a log-concave density that minimises the Kullback--Leibler
divergence from the true density over the class all log-concave densities, and
also show that the log-concave maximum likelihood estimator converges almost
surely in these exponentially weighted total variation norms to this minimiser.
In the case of a correctly specified model, this demonstrates a strong type of
consistency for the estimator; in a misspecified model, it shows that the
estimator converges to the log-concave density that is closest in the
Kullback--Leibler sense to the true density.
