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Mahler Conjecture (non-symmetric case)

Prove that among all convex bodies K ⊂ ℝ^n, the volume product Vol_n(K) · Vol_n(K^∘) is minimized when K is a centered simplex.

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Background

The Mahler conjecture is a central problem in convex geometry concerning the minimum of the volume product. The notes explain that if the simplex maximizes the isotropic constant among all convex bodies, then the Mahler conjecture in the non-symmetric case follows.

The conjecture is known in dimension 2 and is resolved up to an exponential-in-dimension factor via the Bourgain–Milman inequality.

References

the Mahler conjecture suggests that among all convex bodies K ⊂ ℝn, the volume product Vol_n(K) * Vol_n(K{∘}) is minimized when K is a centered simplex [K_adv].

— Isoperimetric inequalities in high-dimensional convex sets (2406.01324 - Klartag et al., 3 Jun 2024) in Section 9 (Bourgain’s slicing problem), bullet list of related conjectures