Large regular simplices contained in a hypercube with a common barycenter

TL;DR

This paper proves that for large dimensions, a regular simplex with a barycenter coinciding with a hypercube can have an edge length exceeding previous bounds, specifically surpassing half the square root of the dimension.

Contribution

It extends prior results by showing larger regular simplices with a common barycenter can be embedded in high-dimensional hypercubes.

Findings

Regular simplices with larger edge lengths exist in high-dimensional hypercubes

The barycenter condition does not limit the maximum size of the simplex

The edge length can exceed 1/2 times the square root of the dimension

Abstract

It has been shown that the $n$-dimensional unit hypercube contains an $n$-dimensional regular simplex of edge length $c\sqrt n$ for arbitrary $c<1/2$ if $n$ is sufficiently large (Maehara, Ruzsa and Tokushige, 2009). We prove the same statement holds for some $c>1/2$ even in the special case where a regular simplex has the same barycenter as that of the unit hypercube.