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 $\mathbb{R}^p$, 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 $\mathcal{M}_{m,n}(\mathbb{R})$, the space of real (m,n)-matrices.