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Topology and geometry of nontrivial rank-one convex hulls for two-by-two matrices

Published online by Cambridge University Press:  22 March 2006

Carl-Friedrich Kreiner
Affiliation:
Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, 04 103 Leipzig, Germany.
Johannes Zimmer
Affiliation:
University of Bath, Department of Mathematical Sciences, Claverton Down, Bath BA2 7AY, UK; zimmer@maths.bath.ac.uk
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Abstract

Continuing earlier work by Székelyhidi, we describe the topological and geometric structure of so-called T4-configurations which are the most prominent examples of nontrivial rank-one convex hulls. It turns out that the structure of T4-configurations in $\mathbb{R}^{2\times 2}$ is very rich; in particular, their collection is open as a subset of $(\mathbb{R}^{2\times 2})^{4}$. Moreover a previously purely algebraic criterion is given a geometric interpretation. As a consequence, we sketch an improved algorithm to detect T4-configurations.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2006

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