We introduce the binacox, a prognostic method to deal with the problem of detect- ing multiple cut-points per features in a multivariate setting where a large number of continuous features are available. The method is based on the Cox model and com- bines one-hot encoding with the binarsity penalty, which uses total-variation regular- ization together with an extra linear constraint, and enables feature selection. Original nonasymptotic oracle inequalities for prediction (in terms of Kullback-Leibler diver- gence) and estimation with a fast rate of convergence are established. The statistical performance of the method is examined in an extensive Monte Carlo simulation study, and then illustrated on three publicly available genetic cancer datasets. On these high- dimensional datasets, our proposed method signi cantly outperforms state-of-the-art survival models regarding risk prediction in terms of the C-index, with a computing time orders of magnitude faster. In addition, it provides powerful interpretability from a clinical perspective by automatically pinpointing signi cant cut-points in relevant variables.