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Semidefinite Representation of the $k$-Ellipse

Jiawang Nie·Pablo A. Parrilo·Bernd Sturmfels
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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 $2^k{-}\binom{k}{k/2}$ 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.

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Jiawang Nie, Pablo A. Parrilo, Bernd Sturmfels. 2007-01-31. Semidefinite Representation of the $k$-Ellipse. https://doi.org/10.1007/978-0-387-75155-9_7

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