Let Ksubseteq R^n be a symmetric convex body, and define its polar body by K^circ=x∈ R^n:langle x,y rangle≤ 1;forall y∈ K. Let B_2^n=x∈ R^n:|x|≤ 1 be the Euclidean unit ball. Ball's conjecture. int_K int_K^circlangle x,y rangle^2,dx,dy≤int_B_2^nint_B_2^nlangle x,y rangle^2,dx,dy. This is a Santaló-type inequality comparing a symmetric convex body with its polar, with the Euclidean ball as the extremal case. The conjecture is resolved by the paper's proof; the abstract also states a stronger isotropic directional inequality and characterizes equality.

Ball's Santaló type inequality for symmetric convex bodies

Ball posed this conjecture in 1986 for symmetric convex bodies $K\subseteq\mathbb{R}^n$ , asserting that the Euclidean ball maximizes the stated polar second-moment product.

Known results

Ball: the conjecture was previously proved for unconditional bodies; the supplied source gives no year for that result.

ArXiv version 3 (date not stated)

Károly J. Böröczky, Konstantinos Patsalos, and Christos Saroglou claim a proof for every symmetric convex body, with equality exactly for origin-symmetric ellipsoids. They also claim a stronger directional inequality under isotropicity, sharp equality for Euclidean balls, and stability consequences for the Blaschke–Santaló inequality. No independent verification, correction, withdrawal, or retraction was found.

Current status (as of September 2026): The conjecture has a published arXiv claim of resolution with an equality characterization, but the proof remains independently unverified.

Ball's Santaló type inequality for symmetric convex bodies

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