Applications of Mathematics, Vol. 62, No. 3, pp. 213-223, 2017
Simplices rarely contain their circumcenter in high dimensions
Jon Eivind Vatne
Received June 27, 2016. First published April 19, 2017.
Abstract: Acute triangles are defined by having all angles less than $\pi/2$, and are characterized as the triangles containing their circumcenter in the interior. For simplices of dimension $n\geq3$, acuteness is defined by demanding that all dihedral angles between $(n-1)$-dimensional faces are smaller than $\pi/2$. However, there are, in a practical sense, too few acute simplices in general. This is unfortunate, since the acuteness property provides good qualitative features for finite element methods. The property of acuteness is logically independent of the property of containing the circumcenter when the dimension is greater than two. In this article, we show that the latter property is also quite rare in higher dimensions. In a natural probability measure on the set of $n$-dimensional simplices, we show that the probability that a uniformly random $n$-simplex contains its circumcenter is $1/2^n$.
Keywords: simplex; circumcenter; finite element method
Affiliations: Jon Eivind Vatne, Department of Computing, Mathematics and Physics, Faculty of Engineering, Western Norway University of Applied Sciences, P. O. Box 7030, N-5020 Bergen, Norway, e-mail: email@example.com