Geometry of multidimensional random walk

Let we have the random walk in d-dimensional space with i.i.d. steps. We are interested in the geometry of the convex hull of the trajectory of this walk. It is discovered that under natural conditions (e.g. if the step distribution has a density) the expectation value of a number of faces of any dimension of the hull does not depend on a step distribution. The talk is based on the joint work with V. Vysotsky and Z. Kabluchko.