We show that the initial cosmological singularity of the universe in the Randall-Sundrum model is a conical singularity of the bulk space-time. Close to this singularity the space-time geometry approaches that of the interior of the future-directed light cone of the singular point. This has two important consequences. First, the problem of the cosmological horizon is naturally resolved in this brane universe model. Secondly, one does not need quantum gravity for the analysis of the problem of initial conditions: the divergence of the four-dimensional curvature of the brane is related to the fact that this quantity is not suitable for the description of light-like surfaces in higher dimensional space-time.