Beitr\ EMIS ELibM Electronic Journals Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 48, No. 1, pp. 225-235 (2007)

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On Reuleaux triangles in Minkowski planes

Horst Martini and Zokhrab Mustafaev

Faculty of Mathematics, University of Technology Chemnitz, 09107 Chemnitz, Germany, e-mail:; Department of Mathematics, University of Houston-Clear Lake, Houston, TX 77058, USA, e-mail:

Abstract: In this paper we prove some results on Reuleaux triangles in (Minkowski or) normed planes. For example, we reprove Wernicke's result (see \cite{We}) that the unit disc and Reuleaux triangles in a normed plane are homothets if and only if the unit circle is either an affine regular hexagon or a parallelogram. Also we show that the ratio of the area of the unit ball of a Minkowski plane to that of a Reuleaux triangle of Minkowski width 1 lies between 4 and 6. The Minkowskian analogue of Barbier's theorem is obtained, and some inequalities on areas of Reuleaux triangles are given.

Keywords: Busemann area, Benson area, Holmes-Thompson area, Minkowski plane, constant Minkowski width, mixed volumes, Reuleaux triangles

Classification (MSC2000): 52A10, 52A40, 46B20, 46B04

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Electronic version published on: 14 May 2007. This page was last modified: 27 Jan 2010.

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