Equipartitions and Mahler volumes of symmetric convex bodies

Following ideas of Iriyeh and Shibata we give a short proof of the three-dimensional Mahler conjecture {\mf for symmetric convex bodies}. Our contributions include, in particular, simple self-contained proofs of their two key statements. The first of these is an equipartition (ham sandwich type) theorem which refines a celebrated result of Hadwiger and, as usual, can be proved using ideas from equivariant topology. The second is an inequality relating the product volume to areas of certain sections and their duals. We observe that these ideas give a large family of convex sets in every dimension for which the Mahler conjecture holds true. Finally we give an alternative proof of the characterization of convex bodies that achieve the equality case and establish a {\mf new} stability result.