Estimates of Quasiconvex Polytopes in the Calculus of Variations

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 rank-one 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.