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