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On the local structure of rank-one convex hulls

Author(s): László Székelyhidi Jr.
Journal: Proc. Amer. Math. Soc. 134 (2006), 1963-1972.
MSC (2000): Primary 26B25
Posted: December 16, 2005
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Abstract: In this note we prove that if $ K$ is a compact set of $ m\times n$ matrices containing an isolated point $ X$ with no rank-one connection into the convex hull of $ K\setminus\{X\}$, then the rank-one convex hull separates as

$\displaystyle K^{rc}=\bigl(K\setminus\{X\}\bigr)^{rc}\cup\{X\}. $

This is an extension of a result of P. Pedregal, which holds for $ 2\times 2$ matrices.

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Additional Information:

László Székelyhidi Jr.
Affiliation: Departement Mathematik, ETH Zentrum, Rämistrasse 101, CH-8092 Zürich, Switzerland
Email: szekelyh@math.ethz.ch

DOI: 10.1090/S0002-9939-05-08299-7
PII: S 0002-9939(05)08299-7
Received by editor(s): February 2, 2005
Posted: December 16, 2005
Additional Notes: The author thanks Bernd Kirchheim for pointing out this problem and for valuable discussions regarding rank-one convexity.
Communicated by: David Preiss
Copyright of article: Copyright 2005, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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