Projective transformations of convex bodies and volume product (Q6624376)

The \textit{volume product} of a convex body \(K\) is the product \(\mbox{vp}(K) := |K||K^\circ|\) of the volumes of \(K\) and its polar body. Among bodies with barycenter at the origin, \(\mbox{vp}(K)\) is maximized only by ellipsoids. The \textit{Mahler conjecture} (proved only for \(d=2\)) says that it's minimized by simplices, with \(\mbox{vp}(\Delta) = (d+1)^{d+1}/(d!)^2\).\N\NKlartag proved a result about covariance matrices in 2018 which implies that a certain strong version of the slicing conjecture implies Mahler's conjecture. Here, the authors give a shorter proof of Klartag's result.