Semidefinite Representation of the $k$-Ellipse

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 $2^k-\left(\genfrac{}{}{0ex}{}{k}{k/2}\right)$ 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.

16 pages, 5 figures

References in corpus (2)

• The Lax conjecture is true
• The Algebraic Degree of Semidefinite Programming

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