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

Issue		ESAIM: COCV Volume 12, Number 2, April 2006


Page(s)		253 - 270
DOI		10.1051/cocv:2005036
Published online		22 March 2006

ESAIM: COCV, April 2006, Vol. 12, pp. 253-270
DOI: 10.1051/cocv:2005036

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

Carl-Friedrich Kreiner¹ and Johannes Zimmer²

¹ Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, 04 103 Leipzig, Germany.
² University of Bath, Department of Mathematical Sciences, Claverton Down, Bath BA2 7AY, UK.

(Received October 14, 2004. / Published online: 22 March 2006)

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.

Mathematics Subject Classification. 49J45, 52A30

Key words: Rank-one convexity, T₄-configurations.

© EDP Sciences, SMAI 2006

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