On perimeters of sections of convex polytopes

In his book “Geometric Tomography” Richard Gardner asks the following question. Let P and Q be origin-symmetric convex bodies in R3 whose sections by any plane through the origin have equal perimeters. Is it true that P=Q? We show that the answer is “Yes” in the class of origin-symmetric convex polytopes. The problem is treated in the general case of Rn.