Improved lower bound for hypercube edge slicing
Lisa Sauermann, Zixuan Xu
TL;DR
The paper proves a new lower bound on the number of hyperplanes needed to slice every edge of the $n$-dimensional hypercube, showing that for large $n$ any slicing hyperplane collection must have size at least $\frac{n^{13/19}}{10^{11}\log^{32/19}n}$. The approach blends a matrix-decomposition strategy with a two-stage randomized construction to place a point $X$ near only a small number of hyperplanes, followed by rounding to a vertex and a probabilistic edge-selection argument framed by anticoncentration theorems. Key technical ingredients include a row/column partition with controlled norms, the notion of vectors containing many scales, and Erdős--Littlewood--Offord-type anticoncentration results to bound the proximity of $X$ to the hyperplanes. The findings tighten the known lower bounds and have implications for depth-two threshold circuits computing parity, highlighting a connection between geometric slicing and circuit complexity. Overall, the work advances understanding of hyperplane coverings and their computational consequences in high dimensions.
Abstract
How many hyperplanes in $\mathbb{R}^n$ are needed in order to slice every edge of the $n$-dimensional hypercube with vertex set $\{\pm 1\}^n$? Here, we say that a hyperplane $H\subseteq \mathbb{R}^n$ slices an edge of the hypercube if it contains exactly one interior point of the edge. The problem of determining the minimum possible size of a collection of hyperplanes in $\mathbb{R}^n$, such that every edge of the hypercube is sliced by at least one of these hyperplanes, is more than 50 years old and has been studied by many researchers. We prove that, for sufficiently large $n$, at least $Ω(n^{13/19}\log^{-32/19}n)$ hyperplanes are needed, improving upon the best previous lower bound $Ω(n^{2/3}\log^{-4/3}n)$ due to Klein.