A Spectral Proof of Khachiyan's Ellipsoid Conjecture
Abstract
For a convex body K⊂ℝn, let w(K) denote the volume of its maximum-volume inscribed ellipsoid. We prove that every closed halfspace H whose boundary passes through the center of the maximizing ellipsoid satisfies w(K∩ H)≤√e/2 w(K). The constant is optimal uniformly over all dimensions, as witnessed by a family of circular cones, thereby establishing Khachiyan's conjecture. The proof converts containment, maximality, and the central-cut condition into algebraic constraints on positive definite matrices. Two complementary spectral bounds from a diagonal model extend to arbitrary center displacements through a directional rank-one estimate for fractional trace powers. Concavity determines their joint optimum. We also derive finite-dimensional bounds and a necessary condition for near equality, with self-contained supporting proofs and an alternative resolvent argument. An AI language model discovered the proof in a human-directed research process. Lean 4 with mathlib verifies the main theorem, sharpness, and supporting results.
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BibTeX
@misc{longfei2026spectral,
title = {{A Spectral Proof of Khachiyan's Ellipsoid Conjecture}},
author = {Zhou Longfei and Haijun Zou and Tianhao Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28447},
eprint = {2609.28447},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28447},
}
RIS
TY - UNPB
TI - A Spectral Proof of Khachiyan's Ellipsoid Conjecture
AU - Zhou Longfei
AU - Haijun Zou
AU - Tianhao Liu
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28447
UR - https://arxiv.org/abs/2609.28447
AB - For a convex body K⊂ℝ^n, let w(K) denote the volume of its maximum-volume inscribed ellipsoid. We prove that every closed halfspace H whose boundary passes through the center of the maximizing ellipsoid satisfies w(K∩ H)≤√e/2 w(K). The constant is optimal uniformly over all dimensions, as witnessed by a family of circular cones, thereby establishing Khachiyan's conjecture. The proof converts containment, maximality, and the central-cut condition into algebraic constraints on positive definite matrices. Two complementary spectral bounds from a diagonal model extend to arbitrary center displacements through a directional rank-one estimate for fractional trace powers. Concavity determines their joint optimum. We also derive finite-dimensional bounds and a necessary condition for near equality, with self-contained supporting proofs and an alternative resolvent argument. An AI language model discovered the proof in a human-directed research process. Lean 4 with mathlib verifies the main theorem, sharpness, and supporting results.
ER -