#### Vol. 8, No. 3, 2015

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On the number of pairwise touching simplices

### Bas Lemmens and Christopher Parsons

Vol. 8 (2015), No. 3, 513–520
##### 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 $n\equiv 1\phantom{\rule{0.2em}{0ex}}mod\phantom{\rule{0.2em}{0ex}}4$. 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.

##### Keywords
touching number, simplices, equilateral sets, $\ell_1$-norm
##### Mathematical Subject Classification 2010
Primary: 52C17
Secondary: 05B40, 46B20
##### Milestones
Received: 17 December 2013
Accepted: 23 February 2014
Published: 5 June 2015

Communicated by Kenneth S. Berenhaut
##### Authors
 Bas Lemmens School of Mathematics, Statistics & Actuarial Science University of Kent Cornwallis Building Canterbury CT2 7NF United Kingdom Christopher Parsons School of Mathematics, Statistics & Actuarial Science University of Kent Cornwallis Building Canterbury CT2 7NF United Kingdom