Simplices rarely contain their circumcenter in high dimensions

• Published in: Applications of mathematics (Prague) Vol. 62; no. 3; pp. 213 - 223
• Main Author: Vatne, Jon Eivind
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2017; Springer Nature B.V
• Subjects: Analysis; Applications of Mathematics; Classical and Continuum Physics; Dimensional measurement; Finite element method; Mathematical analysis; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Optimization; Probability; Theoretical; Topological manifolds; Triangles; simplex; 65M60; finite element method; circumcenter; 52A05
• ISSN: 0862-7940; 1572-9109
• DOI: 10.21136/AM.2017.0187-16

Acute triangles are defined by having all angles less than π/2, and are characterized as the triangles containing their circumcenter in the interior. For simplices of dimension n ≥ 3, acuteness is defined by demanding that all dihedral angles between ( n −1)-dimensional faces are smaller than π/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 .