Title: Smoothness Properties of the Lower Semicontinuous Quasiconvex Envelope
Abstract: Assume that K⊂ℝ\_nm\_ is a convex body with o ∈ int (K) and f\~: ℝ\_nm\_ → ℝ is a Lipschitz resp. C\_1-function. Defining the unbounded function f: ℝ\_nm → ℝ ∪ {(+∞)} through f(v) = {f\~ (ν), ν ∈ K (+∞), v ∈ ℝ\_nm\_ K, we provide sufficient conditions in order to guarantee that its lower semicontinuous quasiconvex envelope f(qc)(w) = sup {g(w) g: ℝ\_nm\_ → ℝ ∪ {(+∞)} quasiconvex and lower semicontinuous, g(ν)≤ f(ν) ∀ ν ∈ ℝ\_nm\_} is globally Lipschitz continuous on K or differentiable in ν ∈ int (K), respectively. An example shows that the partial derivatives of f(qc) do not necessarily admit a representation with a "supporting measure" for fqc in ν0.