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

Kreiner, Carl-Friedrich; Zimmer, Johannes

ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 2, pp. 253-270.

Continuing earlier work by Székelyhidi, we describe the topological and geometric structure of so-called $T_4$-configurations which are the most prominent examples of nontrivial rank-one convex hulls. It turns out that the structure of $T_4$-configurations in ${ℝ}^{2\times 2}$ is very rich; in particular, their collection is open as a subset of ${\left({ℝ}^{2\times 2}\right)}^4$. Moreover a previously purely algebraic criterion is given a geometric interpretation. As a consequence, we sketch an improved algorithm to detect $T_4$-configurations.

Classification : 49J45, 52A30
Mots clés : rank-one convexity, $T_4$-configurations

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