Continuing earlier work by Székelyhidi, we describe the topological and geometric structure of so-called -configurations which are the most prominent examples of nontrivial rank-one convex hulls. It turns out that the structure of -configurations in is very rich; in particular, their collection is open as a subset of . Moreover a previously purely algebraic criterion is given a geometric interpretation. As a consequence, we sketch an improved algorithm to detect -configurations.
Keywords: rank-one convexity, $T_4$-configurations
@article{COCV_2006__12_2_253_0, author = {Kreiner, Carl-Friedrich and Zimmer, Johannes}, 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://www.numdam.org/articles/10.1051/cocv:2005036/} }
TY - JOUR AU - Kreiner, Carl-Friedrich AU - Zimmer, Johannes TI - Topology and geometry of nontrivial rank-one convex hulls for two-by-two matrices JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2006 SP - 253 EP - 270 VL - 12 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2005036/ DO - 10.1051/cocv:2005036 LA - en ID - COCV_2006__12_2_253_0 ER -
%0 Journal Article %A Kreiner, Carl-Friedrich %A Zimmer, Johannes %T Topology and geometry of nontrivial rank-one convex hulls for two-by-two matrices %J ESAIM: Control, Optimisation and Calculus of Variations %D 2006 %P 253-270 %V 12 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2005036/ %R 10.1051/cocv:2005036 %G en %F COCV_2006__12_2_253_0
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, Volume 12 (2006) no. 2, pp. 253-270. doi : 10.1051/cocv:2005036. http://www.numdam.org/articles/10.1051/cocv:2005036/
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