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We revisit the problem of adaptive estimation of the center of symmetry of an unknown symmetric distribution with an additional shape-constraint of log-concavity. This problem was investigated by early authors like Stone (1975), Van Eden (1970), Sacks (1975) who constructed adaptive estimators which depend on tuning parameters. An additional assumption of log-concavity can help us construct simpler estimates which can be efficiently computed without using tuning parameters. To estimate the center of symmetry we consider truncated one-step estimators. To this end, we construct the scores either using the log-concave maximum likelihood estimator (MLE) of the unknown density followed by a symmetrization or by using the symmetric log-concave MLE. We establish that our estimators are $\sqrt{n}-$ consistent with asymptotic efficiency close to 1. These analytical conclusions are supported by simulation studies.