Regularity of quasiconvex envelopes

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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 \(C^{1,\alpha}_{\rm loc}\)) and similar results for other types of envelopes, including polyconvex and rank-1 convex envelopes.