Confexifying the Counting Function on R^(p) for Convexifying the Rank Function on M_(m,n)(R)

We calculate the convex hull of the so-called counting function restricted to a ball of Rp, and we then use it, with a result of A. Lewis, to recover the convex hull of the rank function restricted to a ball of Mm,n(R), the space of real (m,n)-matrices.