  We study nonparametric maximum likelihood estimation of a log-concave density
function $f_0$ which is known to satisfy further constraints, where either (a)
the mode $m$ of $f_0$ is known, or (b) $f_0$ is known to be symmetric about a
fixed point $m$. We develop asymptotic theory for both constrained log-concave
maximum likelihood estimators (MLE's), including consistency, global rates of
convergence, and local limit distribution theory. In both cases, we find the
MLE's pointwise limit distribution at $m$ (either the known mode or the known
center of symmetry) and at a point $x_0 \ne m$. Software to compute the
constrained estimators is available in the R package \verb+logcondens.mode+.
  The symmetry-constrained MLE is particularly useful in contexts of location
estimation. The mode-constrained MLE is useful for mode-regression. The
mode-constrained MLE can also be used to form a likelihood ratio test for the
location of the mode of $f_0$. These problems are studied in separate papers.
In particular, in a separate paper we show that, under a curvature assumption,
the likelihood ratio statistic for the location of the mode can be used for
hypothesis tests or confidence intervals that do not depend on either tuning
parameters or nuisance parameters.
