On Ball's conjectured Santal\'o type inequality

TL;DR

This paper proves a conjecture by Keith Ball regarding a Santaló type inequality for symmetric convex bodies, establishing conditions for equality and providing a stability version of the inequality, thus resolving a notable open problem.

Contribution

It confirms Ball's conjecture on a Santaló type inequality for symmetric convex bodies and introduces a quantitative stability refinement, advancing understanding of convex geometric inequalities.

Findings

Proved Ball's conjecture for symmetric convex bodies.

Established equality conditions for Euclidean balls and ellipsoids.

Developed a stability version of the Blaschke-Santaló inequality.

Abstract

We prove that if $K$ is a symmetric and isotropic convex body in $\mathbb{R}^n$, then $$\int_K\langle x,u\rangle^2\,dx\int_{K^\circ}\langle x,u\rangle^2\,dx\leq \left(\int_{B_2^n}\langle x,u\rangle^2\,dx\right)^2,\qquad\forall u\in\mathbb{R}^n,$$with equality for some $u\neq o$, if and only if $K$ is a Euclidean ball. This confirms a conjecture by Keith Ball (1986), stating that for any symmetric convex body $K$ in $\mathbb{R}^n$, it holds $$\int_K\int_{K^\circ}\langle x,y\rangle^2\,dx\,dy\leq \int_{B_2^n}\int_{B_2^n}\langle x,y\rangle^2\,dx\,dy,$$with equality if and only if $K$ is an ellipsoid. Fortunately, our method for proving Ball's conjectured inequality admits a quantitative stability refinement, which in turn yields an asymptotically optimal stability version of the Blaschke-Santal\'o inequality for origin symmetric convex bodies in terms of the symmetric difference metric. This resolves another well known open problem.