On the maximal perimeter of sections of the cube

We prove that the (n−2)-dimensional surface area (perimeter) of central hyperplane sections of the n-dimensional unit cube is maximal for the hyperplane perpendicular to the vector (1,1,0,…,0). This gives a positive answer to a question of Pełczyński who solved the three dimensional case. We study both the real and the complex versions of this problem. We also use our result to show that the answer to an analogue of the Busemann–Petty problem for the surface area is negative in dimensions 14 and higher.