Local extrema for hypercube sections

TL;DR

This paper investigates the local extrema of the volume of hypercube sections at fixed distances from the center, focusing on hyperplanes orthogonal to diagonals and sub-diagonals, revealing conditions for local maximality across dimensions.

Contribution

It characterizes local extremality of hypercube sections at diagonals and sub-diagonals, providing a comprehensive analysis over the entire range of distances in various dimensions.

Findings

Volume is strictly locally maximal at diagonals for large dimensions within a specific range of t.

At lower order sub-diagonals, volume is locally maximal near t=0 and not extremal for large t.

The characterization allows solving the extremality problem across all feasible t in low to moderate dimensions.

Abstract

Consider the hyperplanes at a fixed distance $t$ from the center of the hypercube $[0,1]^d$. Significant attention has been given to determining the hyperplanes $H$ among these such that the $(d-1)$-dimensional volume of $H\cap[0,1]^d$ is maximal or minimal. In the spirit of a question by Vitali Milman, the corresponding local problem is considered here when $H$ is orthogonal to a diagonal or a sub-diagonal of the hypercube. It is proven in particular that this volume is strictly locally maximal at the diagonals in all dimensions greater than $3$ within a range for $t$ that is asymptotic to $\sqrt{d}/\!\log d$. At lower order sub-diagonals, this volume is shown to be strictly locally maximal when $t$ is close to $0$ and not locally extremal when $t$ is large. This relies on a characterisation of local extremality at the diagonals and sub-diagonals that allows to solve the problem over the whole possible range for $t$ in any fixed, reasonably low dimension.