
Mathematics 2007
Semidefinite Representation of the $k$EllipseDOI: 10.1007/9780387751559_7 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.
