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Zeitschrift für Analysis und ihre Anwendungen


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Volume 29, Issue 4, 2010, pp. 377–400
DOI: 10.4171/ZAA/1413

Published online: 2010-10-02

Smoothness Properties of the Lower Semicontinuous Quasiconvex Envelope

Marcus Wagner[1]

(1) Universität Graz, Austria

Assume that K⊂ℝnm is a convex body with o ∈ int (K) and f~: ℝnm → ℝ is a Lipschitz resp. C1-function. Defining the unbounded function f: ℝnm → ℝ ∪ {(+∞)} through

f(v) = {f~ (ν), ν ∈ K

(+∞), v ∈ ℝnm K,

we provide sufficient conditions in order to guarantee that its lower semicontinuous quasiconvex envelope

f(qc)(w) = sup {g(w) g: ℝnm → ℝ ∪ {(+∞)} quasiconvex and lower semicontinuous, g(ν)≤ f(ν) ∀ ν ∈ ℝnm}

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.

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