Hostname: page-component-848d4c4894-nr4z6 Total loading time: 0 Render date: 2024-06-10T10:40:49.848Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Lie algebras and Lie superalgebras Lie groups

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

[AB]Atiyah, M. F. and Bott, R., ‘The Yang-Mills equations over Riemann surfaces’, Philos. Trans. Roy. Soc. Lond. Ser. A 308 (1982), 523–615.Google Scholar
[BGr]Benson, C. T. and Grove, L. C., Finite reflection groups (Springer, New York, 1985).Google Scholar
[BGe]Berezin, F. A. and Gel'fand, I. M., ‘Some remarks on the theory of spherical functions on symmetric Riemannian manifolds’, Tr. Mosk. Mat. Obshch. 5 (1956), 311–351 (in Russian).Google Scholar
English transl. in Amer. Math. Soc. Transl. 2 21 (1962), 193–238.Google Scholar
[C]Cheng, C. M., ‘An approximation problem with respect to symmetric gauge functions with application to matrix spaces’, Linear and Multilinear Algebra 29 (1991), 169–180.Google Scholar
[H1]Humphreys, J. E., Introduction to Lie algebras and representation theory (Springer, New York, 1972).CrossRefGoogle Scholar
[H2]Humphreys, J. E., Reflection groups and Coxeter groups (Cambridge University Press, Cambridge, 1990).CrossRefGoogle Scholar
[Kn]Knapp, A., Lie groups beyond an introduction, Progress in Math. 140 (Birkhäuser, Boston, 1996).CrossRefGoogle Scholar
[Ko]Kostant, B., ‘On convexity, the Weyl group and Iwasawa decomposition’, Ann. Sci. Ecole Norm. Sup. 6 (1973), 413–460.CrossRefGoogle Scholar
[L]Lewis, A. S., ‘Von Neumann's lemma and a Chevalley-type theorem for convex functions on Cartan subspaces’, Research report (University of Waterloo, 1995).Google Scholar
[LT1]Li, C. K. and Tsing, N. K., ‘Norms that are invariant under unitary similarities and the C-numerical radii’, Linear and Multilinear Algebra 24 (1989), 209–222.Google Scholar
[LT2]Li, C. K. and Tsing, N. K., ‘Distance to the convex hull of the unitary orbit with respect to unitary similarity invariant norms’, Linear and Multilinear Algebra 25 (1989), 93–103.Google Scholar
[MO]Marshall, A. W. and Olkin, I., Inequalities: Theory of majorization and its applications (Academic Press, New York, 1979).Google Scholar
[R]Raîs, M., ‘Remarks on a theorem of R. Westwick’, Technical Report.Google Scholar
[T]Tam, T. Y., ‘A unified extension of two results of Ky Fan on the sum of matrices’, Proc. Amer. Math. Soc. 126 (1998), 2607–2614.Google Scholar
[vN]von Neumann, J., ‘Some matrix-inequalities and metrization ofmatrix-space’, in Collected Works, Vol. 4 (Pergamon, New York, 1962), Tomsk. Univ. Rev. 1 (1937), 286–300.Google Scholar