On the maximal perimeter of sections of the cube

TL;DR

This paper proves that the maximal perimeter of central hyperplane sections of an n-dimensional cube occurs for the hyperplane perpendicular to (1,1,0,...,0), solving a longstanding problem and exploring related geometric questions.

Contribution

It establishes the maximal perimeter configuration for hyperplane sections of the cube and addresses related surface area problems in higher dimensions.

Findings

Maximal perimeter occurs for hyperplanes perpendicular to (1,1,0,...,0)

Answer to Pelczynski's question in the three-dimensional case is confirmed

The Busemann-Petty problem analogue for surface area is negative in dimensions 14 and higher

Abstract

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 Pelczynski 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.