Journal of Geometry

, 110:17 | Cite as

An extremal problem of regular simplices: the five-dimensional case

  • Ákos G. HorváthEmail author
Open Access


The new result of this paper is connected with the following problem: consider a supporting hyperplane of a regular simplex and its reflected image at this hyperplane. When will the volume of the convex hull of these two simplices be maximal? We prove that in the case when the dimension is less or equal to 4, the maximal volume attained in the case when the hyperplane goes through on a vertex and is orthogonal to the height of the simplex at this vertex. More interesting that in the higher dimensional cases this position is not optimal. We also determine an optimal position of the hyperplane in the 5-dimensional case. This corrects an erroneous statement in my paper (Horváth in Beitr Geom Algebra 55(2):415–428, 2014).


Convex hull isometry reflection at a hyperplane simplex volume inequality 

Mathematics Subject Classification

52A40 52A38 26B15 52B11 



Open access funding provided by Budapest University of Technology and Economics (BME). I am thankful to my colleague Zsolt Lángi who found the mistake in the proof of Theorem 3 of paper [3] and inspired me to write the present ones. I also thank for the helpful suggestions of Hans Havlicek and the unknown referee.


  1. 1.
    Filliman, P.: Exterior algebra and projections of polytopes. Discrete Comput. Geom. 5, 305–322 (1990)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Filliman, P.: The extreme projections of the regular simplex. Trans. Am. Math. Soc. 317(2), 611–629 (1990)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Horváth, Á.G.: On an extremal problem connected with simplices. Beitr. Geom. Algebra 55(2), 415–428 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Horváth, Á.G., Lángi, Z.: On the volume of the convex hull of two convex bodies. Monatsh. Math. 174(2), 219–229 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Horváth, Á.G.: Volume of convex hull of two bodies and related problems. In: Conder, M., Deza, A., Weiss, A.I. (eds.) Discrete Geometry and Symmetry. Springer Proceedings in Mathematics and Statistics, vol. 234. Springer, Cham (2018). ISBN: 978-3-319-78433-5CrossRefGoogle Scholar

Copyright information

© The Author(s) 2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of Geometry, Mathematical InstituteBudapest University of Technology and EconomicsBudapestHungary

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