On the number of pairwise touching simplices

Lemmens, Bas; Parsons, Christopher

doi:10.2140/involve.2015.8.513

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On the number of pairwise touching simplices

Bas Lemmens and Christopher Parsons

Vol. 8 (2015), No. 3, 513–520

DOI: 10.2140/involve.2015.8.513

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 $n \geq 5$ such that $n \equiv 1 mod 4$ . The current best known lower bound for general $n$ is $n + 2$ . For $n = 2^{k} - 1$ and $k \geq 2$ , 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

Vol. 8, No. 3, 2015