On Tomaszewski's Cube Vertices Problem

TL;DR

This paper investigates Tomaszewski's cube vertices problem, establishing a lower bound on the number of cube vertices contained within a specific plank intersecting a unit cube, contributing to understanding geometric vertex distributions.

Contribution

It provides a new lower bound for the number of vertices of an n-dimensional cube within a given plank, advancing the geometric understanding of vertex containment.

Findings

Established a lower bound for vertices in the intersection

Connected the problem to a conjecture by Tomaszewski

Enhanced understanding of cube-vertex distribution in geometric planks

Abstract

The following assertion was equivalent to a conjecture proposed by B. Tomaszewski : Let $C$ be an $n$-dimensional unit cube and let $H$ be a plank of thickness $1$, both are centered at the origin, then no matter how to turn the cube around, $C\cap H$ contains at least half of the cube's $2^n$ vertices. A lower bound for the number of the vertices of $C$ in $C\cap H$ was obtained.