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Journal of Convex Analysis 10 (2003), No. 2, 409--417
Copyright Heldermann Verlag 2003

On the Equal Hull Problem for Nontrivial Semiconvex Hulls

Kewei Zhang
School of Mathematical Sciences, University of Sussex, Brighton BN1 9QH, United Kingdom,


We define a nontrivial semiconvex hull $qr_{\alpha}(K)$ of a compact set $K\subset M^{N\times n}$ called the $\alpha$-rank-one convex quadratic hull and establish the equalities of semiconvex hulls with respect to $qr_{\alpha}(K)$ by showing that $L_c(K)=qr_{\alpha}(K)$ if and only if $Q(K)=qr_\alpha(K)$, 0 less than $\alpha$ less than 1, where $Q(K)$ and $L_c(K)$ are the quasiconvex convex hull and the closed lamination convex hull of $K$ respectively. We also show that $qr_{\alpha}(K)$ is a nontrivial semiconvex hull, that is, $qr_{\alpha}(K)\neq C(K)$ if $R(K)\neq C(K)$.

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