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

Kreiner, Carl-Friedrich; Zimmer, Johannes

doi:10.1051/cocv:2005036

Geodesic

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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

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Résumé

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 ${(ℝ^{2 \times 2})}^{4}$ . Moreover a previously purely algebraic criterion is given a geometric interpretation. As a consequence, we sketch an improved algorithm to detect $T_{4}$ -configurations.

MR Zbl

DOI : 10.1051/cocv:2005036

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

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     title = {Topology and geometry of nontrivial rank-one convex hulls for two-by-two matrices},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {253--270},
     publisher = {EDP-Sciences},
     volume = {12},
     number = {2},
     year = {2006},
     doi = {10.1051/cocv:2005036},
     mrnumber = {2209353},
     zbl = {1108.49010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv:2005036/}
}

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Kreiner, Carl-Friedrich; Zimmer, Johannes. Topology and geometry of nontrivial rank-one convex hulls for two-by-two matrices. ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 2, pp. 253-270. doi: 10.1051/cocv:2005036

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