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

Carl Friedrich Kreiner; Johannes Zimmer

doi:10.1051/cocv:2005036

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

Carl Friedrich Kreiner
, Johannes Zimmer

Max Planck Institute for Mathematics in the Sciences
University of Bath, Department of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

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 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.

Original language	English
Pages (from-to)	253-270
Number of pages	18
Journal	ESAIM - Control, Optimisation and Calculus of Variations
Volume	12
Issue number	2
DOIs	https://doi.org/10.1051/cocv:2005036
State	Published - Apr 2006
Externally published	Yes

Keywords

Rank-one convexity
T-configurations

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10.1051/cocv:2005036

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abstract = "Continuing earlier work by Sz{\'e}kelyhidi, 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 ℝ2×2 is very rich; in particular, their collection is open as a subset of (ℝ2×2)4. Moreover a previously purely algebraic criterion is given a geometric interpretation. As a consequence, we sketch an improved algorithm to detect T4-configurations.",

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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 ℝ2×2 is very rich; in particular, their collection is open as a subset of (ℝ2×2)4. Moreover a previously purely algebraic criterion is given a geometric interpretation. As a consequence, we sketch an improved algorithm to detect T4-configurations.

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ER -

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

Abstract

Keywords

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