  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.
