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Distance to the convex hull of an orbit under the action of a compact Lie group

Part of: Lie groups Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  09 April 2009

Randall R. Holmes  and
Tin-Yau Tam
Randall R. Holmes
Affiliation:
Department of Mathematics, Auburn University, Auburn University AL 36849-5310, USA e-mail: holmerr@auburn.edu, tamtiny@auburn.edu
Tin-Yau Tam
Affiliation:
Department of Mathematics, Auburn University, Auburn University AL 36849-5310, USA e-mail: holmerr@auburn.edu, tamtiny@auburn.edu
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Abstract

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For a real vector space V acted on by a group K and fixed x and y in V, we consider the problem of finding the minimum (respectively, maximum) distance, relative to a K-invariant convex function on V, between x and elements of the convex hull of the K-orbit of y. We solve this problem in the case where V is a Euclidean space and K is a finite reflection group acting on V. Then we use this result to obtain an analogous result in the case where K is a maximal compact subgroup of a reductive group G with adjoint action on the vector component ρ of a Cartan decomposition of Lie G. Our results generalize results of Li and Tsing and of Cheng concerning distances to the convex hulls of matrix orbits.

Keywords

Distanceconvex hullorbitreductive Lie group

MSC classification

Secondary: 17B20: Simple, semisimple, reductive (super)algebras 22E46: Semisimple Lie groups and their representations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 66 , Issue 3 , June 1999 , pp. 331 - 357
Copyright
Copyright © Australian Mathematical Society 1999

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