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Monge Points, Euler Lines, and Feuerbach Spheres in Minkowski Spaces

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Discrete Geometry and Symmetry (GSC 2015)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 234))

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Abstract

It is surprising, but an established fact that the field of elementary geometry referring to normed spaces (= Minkowski spaces) is not a systematically developed discipline. There are many natural notions and problems of elementary and classical geometry that were never investigated in this more general framework, although their Euclidean subcases are well known and this extended viewpoint is promising. An example is the geometry of simplices in non-Euclidean normed spaces; not many papers in this direction exist. Inspired by this lack of natural results on Minkowskian simplices, we present a collection of new results as non-Euclidean generalizations of well-known fundamental properties of Euclidean simplices. These results refer to Minkowskian analogues of notions like Euler line, Monge point, and Feuerbach sphere of a simplex in a normed space. In addition, we derive some related results on polygons (instead of triangles) in normed planes.

Dedicated to Károly Bezdek and Egon Schulte on the occasion of their 60th birthdays

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Acknowledgements

The authors are grateful to Emil Molnár for several hints and remarks which helped to improve the presentation in the final version of this paper.

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Authors and Affiliations

  1. Fakultät für Mathematik, Technische Universität Chemnitz, 09107, Chemnitz, Germany

    Undine Leopold & Horst Martini

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  1. Undine Leopold
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  2. Horst Martini
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Correspondence to Horst Martini .

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Editors and Affiliations

  1. Department of Mathematics, University of Auckland, Auckland, New Zealand

    Marston D. E. Conder

  2. Department of Computing and Software, McMaster University, Hamilton, Ontario, Canada

    Antoine Deza

  3. Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada

    Asia Ivić Weiss

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Leopold, U., Martini, H. (2018). Monge Points, Euler Lines, and Feuerbach Spheres in Minkowski Spaces. In: Conder, M., Deza, A., Weiss, A. (eds) Discrete Geometry and Symmetry. GSC 2015. Springer Proceedings in Mathematics & Statistics, vol 234. Springer, Cham. https://doi.org/10.1007/978-3-319-78434-2_13

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