Author

Tianhao Liu

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Preprint Sep 2026

A Curved-Path Refinement of the Kalai--Kleitman Diameter Bound

Let $\Delta_u(d,n)$ be the maximum graph diameter of a pointed $d$-dimensional polyhedron with $n$ facets. Using a curved-path encoding of the iterated Kalai--Kleitman recurrence, we prove, uniformly over $n\geq d\geq 4$, \[ \Delta_u(d,n) \leq (n-d)^{\log_2 G_d},\qquad G_d = (4\ln 2+o(1))\frac{d}{(\ln d)^2}, \] where t...

Tianhao Liu, Dong-Dong Ge, Yin-Yu Ye · 0 citations

Preprint Sep 2026

A Spectral Proof of Khachiyan's Ellipsoid Conjecture

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...

Long-Fei Zhou, Hai-Jun Zou, Tianhao Liu · 0 citations