Mahler’s volume product conjecture

Prove Mahler’s conjecture: determine the minimal value of the volume product P(K) = Vol_d(K)·Vol_d(K°) among centrally symmetric convex bodies K ⊂ R^d and, more generally, establish the minimal volume product over translates for arbitrary convex bodies, showing that it is attained by the cube (symmetric case) and the simplex (general case), respectively, with the stated sharp bounds.

Background

The authors recall Mahler’s classical conjecture from convex geometry to motivate inverse Santaló-type inequalities and their functional analogues discussed in the paper. They also summarize known progress: asymptotic lower bounds, proofs for unconditional bodies, and resolution in small dimensions.

Although this conjecture is classical and not specific to free probability, it underpins the geometric/functional dualities that the paper aims to translate into the free setting.