  We revisit the problem of estimating the center of symmetry $\theta$ of an
unknown symmetric density $f$. Although stone (1975), Eden (1970), and Sacks
(1975) constructed adaptive estimators of $\theta$ in this model, their
estimators depend on external tuning parameters. In an effort to reduce the
burden of tuning parameters, we impose an additional restriction of
log-concavity on $f$. We construct truncated one-step estimators which are
adaptive under the log-concavity assumption. Our simulations suggest that the
untruncated version of the one step estimator, which is tuning parameter free,
is also asymptotically efficient.
  We also study the maximum likelihood estimator (MLE) of $\theta$ in the
shape-restricted model.
