Even log-concave Gaussian-type volume-product conjecture

Prove that for every even log-concave measure \(\mu\) on \(\mathbb{R}^n\) and every centrally symmetric convex body \(K\), the inequality \(\mu(K)\,\mu(K^\circ)\leq \mu(B_2^n)^2\) holds.

Background

The paper studies a Gaussian analogue of the Blaschke–Santaló volume-product problem without assuming that the optimizing set is centrally symmetric. It recalls the known result of Cordero-Erausquin that the Euclidean unit ball maximizes the Gaussian volume product among centrally symmetric convex bodies.

The authors further describe a broader conjecture for arbitrary even log-concave measures and symmetric convex bodies. They note that the conjecture is known for unconditional log-concave measures but remains unresolved in full generality, and that it is connected to the B-conjecture.