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ORBITOPES

Part of: General convexity

Published online by Cambridge University Press:  29 June 2011

Raman Sanyal ,
Frank Sottile  and
Bernd Sturmfels
Raman Sanyal
Affiliation:
Department of Mathematics, University of California, Berkeley, California 94720, U.S.A. (email: sanyal@math.berkeley.edu)
Frank Sottile
Affiliation:
Department of Mathematics, Texas A & M University, College Station, Texas 77843, U.S.A. (email: sottile@math.tamu.edu)
Bernd Sturmfels
Affiliation:
Department of Mathematics, University of California, Berkeley, California 94720, U.S.A. (email: bernd@math.berkeley.edu)
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Abstract

An orbitope is the convex hull of an orbit of a compact group acting linearly on a vector space. These highly symmetric convex bodies lie at the crossroads of several fields, including convex geometry, algebraic geometry, and optimization. We present a self-contained theory of orbitopes, with particular emphasis on instances arising from the groups SO(n) and O(n); these include Schur–Horn orbitopes, tautological orbitopes, Carathéodory orbitopes, Veronese orbitopes, and Grassmann orbitopes. We study their face lattices, algebraic boundaries, and representations as spectrahedra or projected spectrahedra.

MSC classification

Secondary: 14P10: Semialgebraic sets and related spaces 52A05: Convex sets without dimension restrictions 90C22: Semidefinite programming

Information

Type
Research Article
Information
Mathematika , Volume 57 , Issue 2 , July 2011 , pp. 275 - 314
Copyright
Copyright © University College London 2011

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