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Focal Sets of the John Ellipsoid of a Simplex from Its Vertices
Abstract
Starting with the vertices of a simplex in Rn, 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≥2. We give examples in R4, R3, and R2: 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.
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BibTeX
@misc{eydelzon2026focal,
title = {{Focal Sets of the John Ellipsoid of a Simplex from Its Vertices}},
author = {Anatoly Eydelzon},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37448},
eprint = {2609.37448},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37448},
}
RIS
TY - UNPB
TI - Focal Sets of the John Ellipsoid of a Simplex from Its Vertices
AU - Anatoly Eydelzon
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.37448
UR - https://arxiv.org/abs/2609.37448
AB - 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≥2. 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.
ER -