Khachiyan’s Ellipsoid Conjecture

For every integer $n\ge 1$ and every convex body $K\subset\mathbb{R}^n$ , let $E$ be a maximum-volume ellipsoid contained in $K$ , let $c$ be the center of $E$ , and define $w(K)=\operatorname{vol}(E)$ . Then every closed halfspace $H\subset\mathbb{R}^n$ whose boundary contains $c$ satisfies $w(K\cap H)\le \frac{\sqrt{e}}{2}\,w(K)$ . The constant $\frac{\sqrt{e}}{2}$ is optimal uniformly over all dimensions.

References

Primary source

arXiv

Additional references

A Spectral Proof of Khachiyan's Ellipsoid Conjecture — arXiv — Zhou Longfei, Haijun Zou, Tianhao Liu

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed solved

A September 2026 preprint claims to prove the conjecture in every dimension, but the result has not undergone independent mathematical review.

Khachiyan’s conjecture concerns a sharp convex-geometric inequality. The reported optimal constant is $\sqrt{e}/2$ , with equality for circular cones.

September 2026 claimed proof

Zhou Longfei, Haijun Zou, and Tianhao Liu’s preprint A Spectral Proof of Khachiyan's Ellipsoid Conjecture claims the bound uniformly in dimension, identifies circular cones as equality cases, and reports a Lean 4 verification. This is a claimed resolution, not an independently verified one.

Current status (as of September 2026): The conjecture has a preprint claiming a complete proof and formal verification, but its mathematical correctness remains unconfirmed.

Sources

No solutions have been posted yet.