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Angle Measures and Bisectors in Minkowski Planes

Published online by Cambridge University Press:  20 November 2018

Nico Düvelmeyer
Nico Düvelmeyer*
Affiliation:
Technische Universität Chemnitz, Fakultät für Mathematik, D-09107 Chemnitz, Germany e-mail: nico.duevelmeyer@mathematik.tu-chemnitz.de
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Abstract

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We prove that a Minkowski plane is Euclidean if and only if Busemann's or Glogovskij's definitions of angular bisectors coincide with a bisector defined by an angular measure in the sense of Brass. In addition, bisectors defined by the area measure coincide with bisectors defined by the circumference (arc length) measure if and only if the unit circle is an equiframed curve.

Keywords

52A1052A21Radon curvesMinkowski geometryMinkowski planesangular bisectorangular measureequiframed curves
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 48 , Issue 4 , 01 December 2005 , pp. 523 - 534
Copyright
Copyright © Canadian Mathematical Society 2005

References

[1] Averkov, G., On the geometry of simplices in Minkowski spaces. Stud. Univ. Žilina Math. Ser. 16(2003), 1–14.Google Scholar
[2] Brass, P., Erdős distance problems in normed spaces. Comput. Geom. 6(1996), 195–214.Google Scholar
[3] Busemann, H., Planes with analogues to Euclidean angular bisectors. Math. Scand. 36(1975), 5–11.Google Scholar
[4] Düvelmeyer, N., On convex bodies all whose two-dimensional sections are equiframed. Submitted to Arch. Math.Google Scholar
[5] Düvelmeyer, N., A new characterization of Radon curves via angular bisectors. J. Geom. 80(2004), no. 1–2, 75–81.Google Scholar
[6] Glogovs'kiĭ, V. V., Bisectors in the Minkowski plane with norm (xp + yp )1/p. Ukrainian. Vīsnik L'vīv. Polītehn. Īnst. 44, 1970, pp. 192–198, 218.Google Scholar
[7] Helfenstein, H., Problem 13. Canad. Math. Bull. 2(1959), 43.Google Scholar
[8] Helfenstein, H., Solution to problem 13. Canad. Math. Bull. 4 (1961), 77–78.Google Scholar
[9] Martini, H. and Swanepoel, K. J., Equiframed curves—a generalization of Radon curves. Monatsh. Math. 141(2004), 301–314.Google Scholar
[10] Martini, H. and Swanepoel, K. J., The geometry of Minkowski spaces—a survey. II. Expo. Math. 22(2004), 92–144.Google Scholar
[11] Pełczyński, A. and Szarek, S. J., On parallelepipeds of minimal volume containing a convex symmetric body in R n . Math. Proc. Cambridge Philos. Soc. 109(1991), 125–148.Google Scholar
[12] Radon, J., Über eine besondere Art ebener konvexer Kurven, Ber. Sächs. Akad.Wiss. Leipz. 68(1916), 131–134.Google Scholar
[13] Thompson, A. C., Minkowski Geometry, Encyclopedia of Mathematics and its Applications 63, Cambridge University Press, Cambridge, 1996.Google Scholar