On Ball's conjectured Santaló type inequality

Abstract

We prove that if $K$ is a symmetric and isotropic convex body in ${\mathbb{R}}^n$, then $${\int }_K⟨x,u{⟩}^{2}dx{\int }_{K^{\circ }}⟨x,u{⟩}^{2}dx\le {\left({\int }_{B_2^n}⟨x,u{⟩}^{2}dx\right)}^2,\forall u\in {\mathbb{R}}^n,$$with equality for some $u\ne o$, 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 $${\int }_K{\int }_{K^{\circ }}⟨x,y{⟩}^{2}dxdy\le {\int }_{B_2^n}{\int }_{B_2^n}⟨x,y{⟩}^{2}dxdy,$$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-Santaló inequality for origin symmetric convex bodies in terms of the symmetric difference metric. This resolves another well known open problem.

some typos corrected and some proofs simplified