Mahler’s conjecture for convex bodies

Prove that for every convex body K ⊂ ℝ^n, the volume product P(K) = min_{z∈K} vol(K)·vol((K−z)°) satisfies P(K) ≥ P(A_n), where A_n is the n-dimensional regular simplex with barycenter at the origin.

Background

Mahler’s conjecture concerns the minimal value of the volume product of a convex body and its polar, taken at the body’s Santaló point. The Blaschke–Santaló inequality provides the upper bound (attained by the Euclidean ball), while the sharp lower bound is conjectured to be attained by the regular simplex in the non-symmetric setting. The authors recall that, although verified for various families of convex bodies, the general case remains open.

References

The reverse inequality, known as Mahler's conjecture, has been verified for different families of convex bodies (see e.g. [12] and also [22, Section 10.7]), but in general it is still open. This conjecture claims that, for every convex body K, P(K) ≥ P(42) = (n+ 1)2+1 (n!)2 (1.1) , where 4" is a n-dimensional regular simplex with barycenter at the origin.

— Entropy, slicing problem and functional Mahler's conjecture (2406.07406 - Fradelizi et al., 11 Jun 2024) in Section 1 (Introduction), around equation (1.1)