发表机构
Academy of Mathematics and Systems Science, Chinese Academy of Sciences; University of Chinese Academy of Sciences; Antai College of Economics and Management, Shanghai Jiao Tong University（中国科学院数学与系统科学研究院; 中国科学院大学; 上海交通大学安泰经济与管理学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过谱方法证明了Khachiyan关于最大内接椭球中心切割体积比的猜想，给出最优常数√e/2，并由AI辅助发现、Lean 4验证。
AI 中文摘要
对于凸体 $K\subset\mathbb{R}^n$，令 $w(K)$ 表示其最大体积内接椭球的体积。我们证明，任何边界穿过最大椭球中心的闭半空间 $H$ 满足 \\[ w(K\cap H)\le\frac{\sqrt e}{2}\\,w(K)。\\] 该常数在所有维度上一致最优，这由一族圆锥体所证实，从而确立了Khachiyan猜想。证明将包含性、极大性和中心切割条件转化为正定矩阵上的代数约束。来自对角模型的两个互补谱界通过分数迹幂的方向秩一估计推广到任意中心位移。凹性决定了它们的联合最优值。我们还推导了有限维界以及接近相等时的必要条件，并给出了自足的辅助证明和另一种预解式论证。一个AI语言模型在人类指导的研究过程中发现了该证明。Lean 4与mathlib验证了主定理、最优性及辅助结果。
英文摘要
For a convex body $K\subset\mathbb{R}^n$, 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 \[ w(K\cap H)\le\frac{\sqrt e}{2}\,w(K). \] 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.
CommentsLean 4 formalization with mathlib: https://github.com/DrZhouKarl/KhachiyanEllipsoidConjecture
