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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: [email protected])
Frank Sottile
Affiliation:
Department of Mathematics, Texas A & M University, College Station, Texas 77843, U.S.A. (email: [email protected])
Bernd Sturmfels
Affiliation:
Department of Mathematics, University of California, Berkeley, California 94720, U.S.A. (email: [email protected])
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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
Type
Research Article
Information
Mathematika , Volume 57 , Issue 2 , July 2011 , pp. 275 - 314
Copyright
Copyright © University College London 2011

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References

[1]Atiyah, M. F., Convexity and commuting Hamiltonians. Bull. Lond. Math. Soc. 14(1) (1982), 1–15.CrossRefGoogle Scholar
[2]Avis, D., Imai, H. and Ito, T., On the relationship between convex bodies related to correlation experiments with dichotomic observables. J. Phys. A 39(36) (2006), 11283–11299.Google Scholar
[3]Barvinok, A., A Course in Convexity (Graduate Studies in Mathematics 54), American Mathematical Society (Providence, RI, 2002).Google Scholar
[4]Barvinok, A. and Blekherman, G., Convex geometry of orbits. In Combinatorial and Computational Geometry (Mathematical Sciences Research Institute Publications 52), Cambridge University Press (Cambridge, 2005), 51–77.Google Scholar
[5]Barvinok, A. and Novik, I., A centrally symmetric version of the cyclic polytope. Discrete Comput. Geom. 39(1–3) (2008), 76–99.CrossRefGoogle Scholar
[6]Basu, S., Pollack, R. and Roy, M.-F., Algorithms in Real Algebraic Geometry, 2nd edn. (Algorithms and Computation in Mathematics 10), Springer (Berlin, 2006).CrossRefGoogle Scholar
[7]Blekherman, G., Dimensional differences between nonnegative polynomials and sums of squares. Preprint, 2009, arXiv:0907.1339.Google Scholar
[8]Carathéodory, C., Über den Variabilitätsbereich der Koeffizienten von Potenzreihen die gegebene Werte nicht annehmen. Math. Ann. 64 (1907), 95–115.Google Scholar
[9]Dadok, J. and Harvey, R., Calibrations on R 6. Duke Math. J. 50(4) (1983), 1231–1243.Google Scholar
[10]Fan, K., Maximum properties and inequalities for the eigenvalues of completely continuous operators. Proc. Natl. Acad. Sci. USA 37 (1951), 760–766.Google Scholar
[11]Farran, H. R. and Robertson, S. A., Regular convex bodies. J. Lond. Math. Soc. (2) 49(2) (1994), 371–384.CrossRefGoogle Scholar
[12]Fenchel, W., Über Krümmung und Windung geschlossener Raumkurven. Math. Ann. 101(1) (1929), 238–252.CrossRefGoogle Scholar
[13]Fulton, W. and Harris, J., Representation Theory (Graduate Texts in Mathematics 129), Springer (New York, 1991).Google Scholar
[14]Gelfand, I. M., Kapranov, M. M. and Zelevinsky, A. V., Discriminants, Resultants, and Multidimensional Determinants (Mathematics: Theory and Applications), Birkhäuser (Boston, 1994).Google Scholar
[15]Guillemin, V. and Sternberg, S., Convexity properties of the moment mapping. Invent. Math. 67(3) (1982), 491–513.Google Scholar
[16]Harvey, R. and Lawson, H. B. Jr, Calibrated geometries. Acta Math. 148 (1982), 47–157.Google Scholar
[17]Harvey, R. and Morgan, F., The faces of the Grassmannian of three-planes in R 7 (calibrated geometries on R 7). Invent. Math. 83(2) (1986), 191–228.CrossRefGoogle Scholar
[18]Helton, J. W. and Nie, J., Sufficient and necessary conditions for semidefinite representability of convex hulls and sets. SIAM J. Optim. 20(2) (2009), 759–791.CrossRefGoogle Scholar
[19]Henrion, D., Semidefinite representation of convex hulls of rational varieties. Acta Appl. Math. (to appear), arXiv:0901.1821.Google Scholar
[20]Hilbert, D., Über die Darstellung definiter Formen als Summe von Formenquadraten. Math. Ann. 32 (1888), 342–350.CrossRefGoogle Scholar
[21]Lasserre, J. B., Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11(3) (2000/01), 796–817 (electronic).Google Scholar
[22]Leite, R. S., Richa, T. R. W. and Tomei, C., Geometric proofs of some theorems of Schur–Horn type. Linear Algebra Appl. 286(1–3) (1999), 149–173.CrossRefGoogle Scholar
[23]Longinetti, M., Sgheri, L. and Sottile, F., Convex hulls of orbits and orientations of a moving protein domain. Discrete Comput. Geom. 43(1) (2010), 54–77.CrossRefGoogle Scholar
[24]Madden, T. M. and Robertson, S. A., The classification of regular solids. Bull. Lond. Math. Soc. 27(4) (1995), 363–370.Google Scholar
[25]McCarthy, N., Ogilvie, D., Zobin, N. and Zobin, V., Convex geometry of Coxeter-invariant polyhedra. In Trends in Banach Spaces and Operator Theory (Memphis, TN, 2001) (Contemporary Mathematics 321), American Mathematical Society (Providence, RI, 2003), 153–179.Google Scholar
[26]Morgan, F., The exterior algebra ΛkR n and area minimization. Linear Algebra Appl. 66 (1985), 1–28.Google Scholar
[27]Nie, J., Ranestad, K. and Sturmfels, B., The algebraic degree of semidefinite programming. Math. Program. 122 (2010), 379–405.CrossRefGoogle Scholar
[28]Nie, J. and Sturmfels, B., Matrix cubes parametrized by eigenvalues. SIAM J. Matrix Anal. Appl. 31 (2009), 755–766.Google Scholar
[29]Onn, S., Geometry, complexity, and combinatorics of permutation polytopes. J. Combin. Theory Ser. A 64(1) (1993), 31–49.CrossRefGoogle Scholar
[30]Ranestad, K. and Sturmfels, B., The convex hull of a variety. In Notions of Positivity and the Geometry of Polynomials (Trends in Mathematics) (eds Brändén, P., Passare, M. and Putinar, M.), Springer (Basel, 2011), 331–344.CrossRefGoogle Scholar
[31]Recht, B., Fazel, M. and Parrilo, P. A., Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. SIAM Rev. 52(3) (2010), 471–501.Google Scholar
[32]Reznick, B., Sums of even powers of real linear forms. Mem. Amer. Math. Soc. 96(463) (1992).Google Scholar
[33]Schur, I., Über eine Klasse von Mittelbildungen mit Anwendungen auf die Determinantentheorie. Theorie Sitzungsber. Berlin Math. Ges. 22 (1923), 9–20.Google Scholar
[34]Smilansky, Z., Convex hulls of generalized moment curves. Israel J. Math. 52(1–2) (1985), 115–128.Google Scholar
[35]Sturmfels, B. and Uhler, C., Multivariate Gaussians, semidefinite matrix completion and convex algebraic geometry. Ann. Inst. Statist. Math. 62 (2010), 603–638.Google Scholar
[36]Vinzant, C., Edges of the Barvinok–Novik orbitope. Discrete Comput. Geom. (2011), doi:10.1007/S00454-011-9351-y, arXiv:1003.4528 .CrossRefGoogle Scholar
[37]Weis, S., A note on touching cones and faces. J. Convex Anal. 19(2) (2012), available online athttp://www.heldermann.de/JCA/JCA19/JCA192/jca19018.htm.Google Scholar
[38]Ziegler, G. M., Lectures on Polytopes (Graduate Texts in Mathematics 152), Springer (New York, 1995).Google Scholar