On Ball's conjectured Santaló type inequality
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-Santaló inequality for origin symmetric convex bodies in terms of the symmetric difference metric. This resolves another well known open problem.