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Semidefinite Representation of the k-Ellipse

  • Jiawang Nie
  • Pablo A. Parrilo
  • Bernd Sturmfels
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 146)

Abstract

The k-ellipse is the plane algebraic curve consisting of all points whose sum of distances from k given points is a fixed number. The polynomial equation defining the k-ellipse has degree 2 k if k is odd and degree \( 2^k - \left( {\begin{array}{*{20}c} k \\ {k/2} \\ \end{array} } \right) \) if k is even. We express this polynomial equation as the determinant of a symmetric matrix of linear polynomials. Our representation extends to weighted k-ellipses and k-ellipsoids in arbitrary dimensions, and it leads to new geometric applications of semidefinite programming.

Key words

k-ellipse algebraic degree semidefinite representation Zariski closure tensor sum 

AMS(MOS) subject classifications

90C22 52A20 14Q10 

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References

  1. [1]
    C. BAJAJ. The algebraic degree of geometric optimization problems. Discrete Comput. Geom., 3(2):177–191, 1988.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    R. BELLMAN. Introduction to Matrix Analysis. Society for Industrial and Applied Mathematics (SIAM), 1997.Google Scholar
  3. [3]
    R.R. CHANDRASEKARAN AND A. TAMIR. Algebraic optimization: the Fermat-Weber location problem. Math. Programming, 46(2, (Ser. A)):219–224, 1990.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    M.R. GAREY AND D.S. JOHNSON. Computers and Intractability: A guide to the theory of NP-completeness. W.H. Freeman and Company, 1979.Google Scholar
  5. [5]
    J.W. HELTON AND V. VINNIKOV. Linear matrix inequality representation of sets. To appear in Comm. Pure Appl. Math. Preprint available from arxiv.org/ abs/math.OC/0306180. 2003.Google Scholar
  6. [6]
    R.A. HORN AND C.R. JOHNSON. Topics in Matrix Analysis. Cambridge University Press, 1994.Google Scholar
  7. [7]
    D.K. KULSHRESTHA. k-elliptic optimization for locational problems under constraints. Operational Research Quarterly, 28(4-l):871–879, 1977.zbMATHCrossRefGoogle Scholar
  8. [8]
    A.S. LEWIS, P.A. PARRILO, AND M.V. RAMANA. The Lax conjecture is true. Proc. Amer. Math. Soc., 133(9):2495–2499, 2005.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    M. LOBO, L. VANDENBERGHE, S. BOYD, AND H. LEBRET. Applications of second-order cone programming. Linear Algebra and its Applications, 284:193–228, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    T. MIGNON AND N. RESSAYRE. A quadratic bound for the determinant and permanent problem. International Mathematics Research Notices, 79:4241–4253, 2004.CrossRefMathSciNetGoogle Scholar
  11. [11]
    G. SZ.-NAGY. Tschirnhaussche Eiflächen und Eikurven. Acta Math. Acad. Sci. Hung. 1:36–45, 1950.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    J. NIE, K. RANESTAD, AND B. STURMFELS. The algebraic degree of semidefinite programming. Preprint, 2006, math.0C/0611562.Google Scholar
  13. [13]
    J. RENEGAR. Hyperbolic programs, and their derivative relaxations. Found. Comput. Math. 6(1):59–79, 2006.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    J. SEKINO. n-ellipses and the minimum distance sum problem. Amer. Math. Monthly, 106(3): 193–202, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    H. STURM. Über den Punkt kleinster Entfernungssumme von gegebenen Punkten. Journal für die Reine und Angewandte Mathematik 97:49–61, 1884.CrossRefGoogle Scholar
  16. [16]
    C.M. TRAUB. Topological Effects Related to Minimum Weight Steiner Triangulations. PhD thesis, Washington University, 2006.Google Scholar
  17. [17]
    L. VANDENBERGHE AND S. BOYD. Semidefinite programming. SIAM Review, 38: 49–95, 1996.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    E.V. WEISZFELD. Sur le point pour lequel la somme des distances de n points donnés est minimum. Tohoku Mathematical Journal 43:355–386, 1937.Google Scholar
  19. [19]
    H. WOLKOWICZ, R. SAIGAL, AND L. VANDENBERGHE (Eds.). Handbook of Semidefinite Programming. Theory, Algorithms, and Applications Series: International Series in Operations Research and Management Science, Vol. 27, Springer Verlag, 2000.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Jiawang Nie
    • 1
  • Pablo A. Parrilo
    • 2
  • Bernd Sturmfels
    • 3
  1. 1.Department of MathematicsUniversity of California at San DiegoLa Jolla
  2. 2.Laboratory for Information and Decision SystemsMassachusetts Institute of TechnologyCambridge
  3. 3.Department of MathematicsUniversity of California at BerkeleyBerkeley

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