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Zbl 1038.26008
Kolář, Jan
Non-compact lamination convex hulls.
(English)
[J] Ann. Inst. Henri Poincaré, Anal. Non Linéaire 20, No. 3, 391-403 (2003). ISSN 0294-1449

The lamination convex hull of a set $K$ of $m\times n$ real matrices is the smallest set $L(K)$ that contains $K$ and has the property that $(1-\lambda)A+\lambda B\in L(K)$ whenever $A, B\in K$, $\text{rank}(A-B) = 1$ and $\lambda \in (0,1).$ The author proves by means of an example that a compact set $K$ of symmetric $2\times 2$ matrices need not have a compact lamination convex hull. A similar result is proved for separately lamination convex hulls in $\Bbb R^3$ (the separately lamination convex hull of $K\subset \Bbb R^n$ is the smallest set containing $K$ and every segment with endpoints in $K$ which is parallel to one of the coordinate axis) and for bi-convex hulls in $\Bbb R^2\times \Bbb R$ (the bi-convex hull of $A\subset \Bbb R^k\times \Bbb R^l$ is the smallest set with convex sections that contains $A$). It is also shown that the mapping assigning to $K$ the closure of $L(K)$ is not upper semicontinuous on the space of diagonal $3\times 3$ matrices with respect to the Hausdorff metric.
[Juan-Enrique Mart\'inez-Legaz (Barcelona)]
MSC 2000:
*26B25 Convexity and generalizations (several real variables)
52A30 Variants of convex sets
15A99 Miscellaneous topics in linear algebra
49J45 Optimal control problems inv. semicontinuity and convergence
90C25 Convex programming

Keywords: lamination convexity; separate convexity; bi-convexity; upper semicontinuity; convex hull

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