  In this article, we revisit the problem of estimating the unknown
zero-symmetric distribution in a two-component location mixture model,
considered in previous works, now under the assumption that the zero-symmetric
distribution has a log-concave density. When consistent estimators for the
shift locations and mixing probability are used, we show that the nonparametric
log-concave Maximum Likelihood estimator (MLE) of both the mixed density and
that of the unknown zero-symmetric component are consistent in the Hellinger
distance. In case the estimators for the shift locations and mixing probability
are $\sqrt n$-consistent, we establish that these MLE's converge to the truth
at the rate $n^{-2/5}$ in the $L_1$ distance. To estimate the shift locations
and mixing probability, we use the estimators proposed by
\cite{hunteretal2007}. The unknown zero-symmetric density is efficiently
computed using the \proglang{R} package \pkg{logcondens.mode}.
