A Spectral Proof of Khachiyan's Ellipsoid Conjecture

Abstract

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.