The submission provides a polynomial-time approximation algorithm for finding the maximum-likelihood log-concave density for a given set of data points in R^d, for arbitrary d. The work is theoretical in nature, with proofs and no experiments. The problem is very interesting, since log-concave distributions include may of the commonly used parametric families (such as Gaussian), and the log-concave MLE has also other interesting properties. Previously the sample-complexity of learning a log-concave distribution has been studied, but a polynomial-time algorithm has been lacking. The present work provides such an algorithm. The derivation of the algorithm requires solving several interesting and non-trivial technical sub-problems. The presentation is generally very good. On slightly negative side, the practical impact of the result in very high-dimensional settings might be limited because of the high degree of the polynomial dependence on d.