Volumes of complementary projections of convex polytopes (Q1092414)

It is shown that those centrally symmetric convex d-polytopes whose images under orthogonal projection on to any pair of orthogonal complementary subspaces of \({\mathbb{E}}^ d\) have numerically equal volumes are cartesian products of (centrally symmetric) polygons in \({\mathbb{E}}^ 2\) with the same property and unit line segments. For \(d\geq 3\), the general projection property in fact follows from that for pairs of hyperplanes and lines. A conjecture is made about the solution of the problem in the non-centrally symmetric case.