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

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

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.

Keywords

Birkhoff orthogonality Vertex centroid Circumsphere Euler line Feuerbach sphere Isosceles orthogonality Mannheim’s theorem Minkowskian simplex Monge point Normality Normed space 

2010 Mathematics Subject Classification

46B20 51M05 51M20 52A10 52A20 52A21 52B11 

Notes

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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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Fakultät für MathematikTechnische Universität ChemnitzChemnitzGermany

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