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 $2^k{}\binom{k}{k/2}$ 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 January 2007
 arXiv:
 arXiv:math/0702005
 Bibcode:
 2007math......2005N
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Optimization and Control
 EPrint:
 16 pages, 5 figures