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Upper semicontinuity of the lamination hull

Terence L.J. Harris

2018
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E S A I M: Control, Optimisation and Calculus of Variations
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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 Kol\'a\v{r}. 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,
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... set $K$ is given, which has $L(K)$ connected, compact and strictly smaller than $K^{rc}$.

doi:10.1051/cocv/2017033
fatcat:b3uvtvnorrftdoipcxi37qlg4e