Focal Sets of the John Ellipsoid of a Simplex from Its Vertices

Abstract

Starting with the vertices of a simplex in $\R^n$, we build a cubic vector field. If the semiaxes of the simplex's John ellipsoid are distinct, the Jacobian of this field has repeated eigenvalues exactly on the ellipsoid's focal sets. This works in every dimension $n\geq2$. We give examples in $\R^4$, $\R^3$, and $\R^2$: three focal quadrics, an ellipse and a hyperbola, and finally two points. In the plane, these two points are also the zeros of the derivative in the Siebeck--Marden theorem. Complex numbers explain why the planar case has a shorter formula.