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 2k if k is odd and degree
2k−(k/2k​) 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.Comment: 16 pages, 5 figure