
Journal of Convex Analysis 13 (2006), No. 1, 037050 Copyright Heldermann Verlag 2006 Estimates of Quasiconvex Polytopes in the Calculus of Variations Kewei Zhang Dept. of Mathematics, University of Sussex, Falmer, Brighton BN1 9RF, Great Britain k.zhang@sussex.ac.uk [Abstractpdf] \def\dist{\operatorname{dist}} We give direct estimates for the quasiconvex polytopes $Q(K)$ generated by a finite set $K\subset M^{N\times n}$. More precisely, we bound the quasiconvex envelope $Q\dist(\cdot,K)$ near a convex exposed face of $C(X)$ which does not have rankone connections. Our estimates depend on the weak(1,1) bounds for certain singular integral operators and the geometric features of the convex polytope $C(K)$. We show by an example that our estimate is `local' and independent of the `size' of $K$, hence it is a better estimate than the polyconvex hull $P(K)$ which is `size' dependent. Keywords: Direct estimates, quasiconvex polytopes, quasiconvex envelope, singular integral operators, polyconvex hull. [ Fulltextpdf (412 KB)] for subscribers only. 