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)


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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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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