# Smoothness Properties of the Lower Semicontinuous Quasiconvex Envelope

### Marcus Wagner

Universität Graz, Austria

## 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.

## Cite this article

Marcus Wagner, Smoothness Properties of the Lower Semicontinuous Quasiconvex Envelope. Z. Anal. Anwend. 29 (2010), no. 4, pp. 377–400

DOI 10.4171/ZAA/1413