Applications of Mathematics, online first, 11 pp.

Simplices rarely contain their circumcenter in high dimensions

Jon Eivind Vatne

Received June 27, 2016.   First published April 19, 2017.

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:

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

Classification (MSC 2010): 65M60, 52A05

DOI: 10.21136/AM.2017.0187-16

Full text available as PDF.

  [1] J. Bertrand: Calcul des probabilités. Gauthier-Villars, Paris (1889). JFM 21.0198.01
  [2] J. Brandts, S. Korotov, M. Křížek: Dissection of the path-simplex in $\Bbb R^n$ into $n$ path-subsimplices. Linear Algebra Appl. 421 (2007), 382-393. DOI 10.1016/j.laa.2006.10.010 | MR 2294350 | Zbl 1112.51006
  [3] 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), 103-122. DOI 10.2174/978160805291211101010103
  [4] P. G. Ciarlet: Basic error estimates for elliptic problems. Handbook of Numerical Analysis. Volume II: Finite Element Methods (Part 1). North-Holland, Amsterdam (1991), 17-351. DOI 10.1016/s1570-8659(05)80039-0 | MR 1115237 | Zbl 0875.65086
  [5] M. Hajja, P. Walker: Equifacial tetrahedra. Int. J. Math. Educ. Sci. Technol. 32 (2001), 501-508. DOI 10.1080/00207390110038231 | MR 1847966 | Zbl 1011.51007
  [6] R. Hošek: Face-to-face partition of 3D space with identical well-centered tetrahedra. Appl. Math., Praha 60 (2015), 637-651. DOI 10.1007/s10492-015-0115-5 | MR 3436566 | Zbl 06537666
  [7] G. Kalai: On low-dimensional faces that high-dimensional polytopes must have. Combinatorica 10 (1990), 271-280. DOI 10.1007/BF02122781 | MR 1092544 | Zbl 0721.52008
  [8] E. Kopczyński, I. Pak, P. Przytycki: Acute triangulations of polyhedra and $\mathbb{R}^N$. Combinatorica 32 (2012), 85-110. DOI 10.1007/s00493-012-2691-2 | MR 2927633 | Zbl 1265.52014
  [9] S. Korotov, M. Křížek: Global and local refinement techniques yielding nonobtuse tetrahedral partitions. Comput. Math. Appl. 50 (2005), 1105-1113. DOI 10.1016/j.camwa.2005.08.012 | MR 2167747 | Zbl 1086.65116
  [10] S. Korotov, J. Stańdo: Yellow-red and nonobtuse refinements of planar triangulations. Math. Notes, Miskolc 3 (2002), 39-46. MR 1921485 | Zbl 1017.65086
  [11] M. Křížek: On the maximum angle condition for linear tetrahedral elements. SIAM J. Numer. Anal. 29 (1992), 513-520. DOI 10.1137/0729031 | MR 1154279 | Zbl 0755.41003
  [12] M. Křížek: There is no face-to-face partition of $\bold R^5$ into acute simplices. Discrete Comput. Geom. 36 (2006), 381-390; Erratum. Discrete Comput. Geom. 44 (2010), 484-485 . DOI 10.1007/s00454-006-1244-0 | MR 2252110 | Zbl 1103.52008
  [13] M. E. Muller: A note on a method for generating points uniformly on $N$-dimensional spheres. Commun. ACM 2 (1959), 19-20. DOI 10.1145/377939.377946 | Zbl 0086.11605
  [14] E. VanderZee, A. N. Hirani, D. Guoy: Triangulation of simple 3D shapes with well-centered tetrahedra. Proceedings of the 17th International Meshing Roundtable. Springer, Berlin (2008), 19-35. DOI 10.1007/978-3-540-87921-3_2
  [15] E. VanderZee, A. N. Hirani, V. Zharnitsky, D. Guoy: A dihedral acute triangulation of the cube. Comput. Geom. 43 (2010), 445-452. DOI 10.1016/j.comgeo.2009.09.001 | MR 2585566 | Zbl 1185.65040

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