On Tomaszewski's Cube Vertices Problem

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.