Upper semicontinuity of the lamination hull

Abstract

Let $K\subseteq {\mathbb{R}}^{2\times 2}$ be a compact set, let $K^{rc}$ be its rank-one convex hull, and let $L(K)$ be its lamination convex hull. It is shown that the mapping $K\to \overline{L(K)}$ is not upper semicontinuous on the diagonal matrices in ${\mathbb{R}}^{2\times 2}$, which was a problem left by Kolář. This is followed by an example of a 5-point set of $2\times 2$ symmetric matrices with non-compact lamination hull. Finally, an example of another 5-point set $K$ is given, which has $L(K)$ connected, compact and strictly smaller than $K^{rc}$.

arXiv comment8 pages, 2 figures. Accepted version