From the Steiner Inellipse to the John Ellipsoid of a Simplex: A Corner-Volume Characterization
Abstract
For a triangle of area T, a planar corner-area characterization states that an interior point M lies on the Steiner inellipse precisely when the three corner triangles cut off by the lines through M parallel to the sides have areas T1,T2,T3 satisfying T1+T2+T3=1/2 T. We give the corresponding statement for a simplex in arbitrary dimension. If S is a nondegenerate n-simplex of volume V and V1(M),…,Vn+1(M) are the volumes of the facet-parallel corner simplices determined by M, then M∈∂ EJ(S) Longleftrightarrow ∑i=1n+1Vi(M)2/n=1/n V2/n, where EJ(S) is the John ellipsoid of S. We also identify the entire corner-volume functional with the central second-moment quadratic of the uniform simplex. The novelty claimed here is limited to the corner-volume formulations and their connections with the planar Steiner-inellipse result; the underlying barycentric, covariance, and John-ellipsoid facts are classical.
Related
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- The Ekeland--Hofer--Zehnder Capacity of rotated Lp-Ellipsoid
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BibTeX
@misc{eydelzon2026from,
title = {{From the Steiner Inellipse to the John Ellipsoid of a Simplex: A Corner-Volume Characterization}},
author = {Anatoly Eydelzon},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08045},
eprint = {2609.08045},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08045},
}
RIS
TY - UNPB
TI - From the Steiner Inellipse to the John Ellipsoid of a Simplex: A Corner-Volume Characterization
AU - Anatoly Eydelzon
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.08045
UR - https://arxiv.org/abs/2609.08045
AB - For a triangle of area T, a planar corner-area characterization states that an interior point M lies on the Steiner inellipse precisely when the three corner triangles cut off by the lines through M parallel to the sides have areas T_1,T_2,T_3 satisfying T_1+T_2+T_3=1/2 T. We give the corresponding statement for a simplex in arbitrary dimension. If S is a nondegenerate n-simplex of volume V and V_1(M),…,V_n+1(M) are the volumes of the facet-parallel corner simplices determined by M, then M∈∂ E_J(S) Longleftrightarrow ∑_i=1^n+1V_i(M)^2/n=1/n V^2/n, where E_J(S) is the John ellipsoid of S. We also identify the entire corner-volume functional with the central second-moment quadratic of the uniform simplex. The novelty claimed here is limited to the corner-volume formulations and their connections with the planar Steiner-inellipse result; the underlying barycentric, covariance, and John-ellipsoid facts are classical.
ER -