On the number of pairwise touching simplices

Abstract

In this note, it is shown that the maximum number of pairwise touching translates of an $n$-simplex is at least $n+3$ for $n=7$, and for all $n\ge 5$ such that $n\equiv 1mod4$. The current best known lower bound for general $n$ is $n+2$. For $n=2^k-1$ and $k\ge 2$, we will also present an alternative construction to give $n+2$ touching simplices using Hadamard matrices.