Journal of Convex Analysis, Vol. 5, No. 1, pp. 133-146 (1998)

On Some Quasiconvex Functions with Linear Growth

Kewei Zhang

Department of Mathematics, Macquarie University, North Ryde, Sydney 2109, Australia,

Abstract: We establish (i) that the quasiconvexifcation of the distance function to any closed (possibly unbounded) subset of the space of conformal matrices $E_{\partial}$ in $M^{2\times 2}$ is bounded from below by the distance function itself, that is, $Q \operatorname{dist}(\cdot,K)\geq c \operatorname{dist}(\cdot,K)$, where $c>0$ is a constant independent of $K$; (ii) some estimates of quasiconvexifications of the distance function to a closed subset of $M^{2\times 2}$ which is `supported' by $E_\partial$; (iii) $Q\operatorname{dist}^p(\cdot,K)=Q\dist^p(\cdot,Q_p(K))$ for any $p\geq 1$ and any closed $K\subset M^{N\times n}$; (iv) for some nonconvex $K\subset M^{2\times 2}$, $Q\operatorname{dist}(\cdot, K)$ is homogeneous of degree one, conjugate invariant and convex, and $Q_1(K)=C(K)$.

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