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 5 such that n 1 mod 4. The current best known lower bound for general n is n + 2. For n = 2k 1 and k 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