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Zeitschrift für Analysis und ihre Anwendungen

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Published online: 2010-10-02

Smoothness Properties of the Lower Semicontinuous Quasiconvex Envelope

Marcus Wagner^[1]

(1) Universität Graz, Austria

Assume that K⊂ℝ^nm is a convex body with o ∈ int (K) and f^~: ℝ^nm → ℝ is a Lipschitz resp. C¹-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 f^qc in ν₀.

Keywords: Quasiconvex function, separately convex function, lower semicontinuous quasiconvex envelope, probability measure, differentiability

Wagner Marcus: Smoothness Properties of the Lower Semicontinuous Quasiconvex Envelope. Z. Anal. Anwend. 29 (2010), 377-400. doi: 10.4171/ZAA/1413