# 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

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topKreiner, 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 -

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