Mahler's conjecture for some hyperplane sections (Q2022798)

Mahler's conjecture is one of the most important open problems in Convexity: in 1939 Mahler conjectured that for every centrally symmetric convex body \(K\subset{\mathbb R}^n\), the product of its volume and the volume of its polar satisfies the inequality \[ \mathrm{vol}(K)\mathrm{vol}(K^{\circ})\geq\frac{4^n}{n!}. \] Mahler himself confirmed the conjecture in the plane, whereas the \(3\)-dimen\-sio\-nal case has been recently established in [\textit{H. Iriyeh} and \textit{M. Shibata}, Duke Math. J. 169, 1077--1134 (2020; Zbl 1439.52007)]. Many other partial results have appeared over the years. The cube and the cross-polytope, optimal sets in the above inequality, are particular cases of the so-called Hanner polytopes, which have been recently proved to be local minima for the volume product (see [\textit{F. Nazarov} et al., Duke Math. J. 154, 419--430 (2010; Zbl 1207.52005); \textit{J. Kim}, J. Funct. Anal. 266, 2360--2402 (2014; Zbl 1361.52003)]). Hanner polytopes are defined as those centrally symmetric polytopes which are obtained from segments by repeatedly applying one of the following operations: taking either the Cartesian product or the \(\ell_1\)-sum. In the paper under review the author shows, using symplectic techniques, that Mahler's conjecture also holds for hyperplane sections (and projections) of \(\ell_p\)-balls, \(1\leq p\leq+\infty\), and of Hanner polytopes.