On the Equal Hull Problem for Nontrivial Semiconvex Hulls

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 $Q(K)=qr_\alpha(K)$, $0<\alpha<1$, 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 $R(K)\neq C(K)$