Equipartitions and Mahler volumes of symmetric convex bodies

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