Regularity of quasiconvex envelopes

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Quasiconvex functions incorporating volumetric constraints are rank-one convex (2008, background): “For higher regularity of quasiconvex envelopes see [ 4 ].”
Progress and puzzles in nonlinear elasticity (2010, background): “It is not difficult to show that W ∗∗ is C1 (for a general result of this type see Kirchheim & Kristensen [20]).”
Weak Solutions for a Class of Nonlinear Systems of Viscoelasticity (2000, background): “theorem of Ball, Kirchheim & Kristensen [ 8 ] which establishes that the differentiability properties of a function are inherited, and in fact improved, by its quasiconvex envelope.”
On Rank One Convex Functions that are Homogeneous of Degree One (2015, result): “ted component of A ∩ ℓcontaining x. The result follows since we can connect any point y∈ A with xby such line segments contained in A. The following result and its proof is a variant of a result from [BKK00] (compare Lemma 2.2 there) that in turn is a slightly more precise version of a well–known es”