Convexifying the set of matrices of bounded rank: applications to the quasiconvexification and convexification of the rank function

We provide an explicit description of the convex hull of the set of matrices of bounded rank, restricted to balls for the spectral norm. As applications, we deduce two relaxed forms of the rank function restricted to balls for the spectral norm: one is the quasiconvex hull of this rank function, another one is the convex hull of the rank function, thus retrieving Fazel's theorem (2002). Key-words: Rank of a matrix; spectral norm; trace (or nuclear) norm; quasiconvex hull of a function; convex hull of a set; convex hull of a function.