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

, 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}) M m , n ( R )

, the space of real (m,n)-matrices.