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

Carl-Friedrich Kreiner; Johannes Zimmer

ESAIM: Control, Optimisation and Calculus of Variations (2006)

  • Volume: 12, Issue: 2, page 253-270
  • ISSN: 1292-8119

Abstract

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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 ℝ 2 × 2 is very rich; in particular, their collection is open as a subset of ( ℝ 2 × 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.

How to cite

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Kreiner, Carl-Friedrich, and Zimmer, Johannes. "Topology and geometry of nontrivial rank-one convex hulls for two-by-two matrices." ESAIM: Control, Optimisation and Calculus of Variations 12.2 (2006): 253-270. <http://eudml.org/doc/249663>.

@article{Kreiner2006,
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. },
author = {Kreiner, Carl-Friedrich, Zimmer, Johannes},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Rank-one convexity; T4-configurations.; -configurations},
language = {eng},
month = {3},
number = {2},
pages = {253-270},
publisher = {EDP Sciences},
title = {Topology and geometry of nontrivial rank-one convex hulls for two-by-two matrices},
url = {http://eudml.org/doc/249663},
volume = {12},
year = {2006},
}

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
DA - 2006/3//
PB - EDP Sciences
VL - 12
IS - 2
SP - 253
EP - 270
AB - 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.
LA - eng
KW - Rank-one convexity; T4-configurations.; -configurations
UR - http://eudml.org/doc/249663
ER -

References

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