Simplices rarely contain their circumcenter in high dimensions
- First Online:
- 41 Downloads
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 1/2n.
Keywordssimplex circumcenter finite element method
MSC 201065M60 52A05
Unable to display preview. Download preview PDF.
- J. Brandts, S. Korotov, M. Křížek: A geometric toolbox for tetrahedral finite element partitions. Efficient Preconditioned Solution Methods for Elliptic Partial Differential Equations (O. Axelsson, J. Karátson, eds.). Bentham Science Publishers Ltd., 2011, pp. 103–122.Google Scholar
- P. G. Ciarlet: Basic error estimates for elliptic problems, Handbook of Numerical Analysis. Volume II: Finite Element Methods (Part 1). North-Holland, Amsterdam, 1991, pp. 17–351.Google Scholar
- E. VanderZee, A. N. Hirani, D. Guoy: Triangulation of simple 3D shapes with wellcentered tetrahedra, Proceedings of the 17th International Meshing Roundtable. Springer, Berlin, 2008, pp. 19–35.Google Scholar