Regularity of quasiconvex envelopes

We prove that the quasiconvex envelope of a differentiable function which satisfies natural growth conditions at infinity is a C-1 function. Without the growth conditions the result fails in general. We also obtain results on higher regularity (in the sense of C-loc(1,alpha)) and similar results for other types of envelopes, including polyconvex and rank-1 convex envelopes.