Geodesic
Angle Measures and Bisectors in Minkowski Planes
Canadian mathematical bulletin, Tome 48 (2005) no. 4, pp. 523-534

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CMB-2005-048-0
Mots-clés : 52A10, 52A21, Radon curves, Minkowski geometry, Minkowski planes, angular bisector, angular measure, equiframed curves 
Düvelmeyer, Nico. Angle Measures and Bisectors in Minkowski Planes. Canadian mathematical bulletin, Tome 48 (2005) no. 4, pp. 523-534. doi: 10.4153/CMB-2005-048-0
@article{10_4153_CMB_2005_048_0,
     author = {D\"uvelmeyer, Nico},
     title = {Angle {Measures} and {Bisectors} in {Minkowski} {Planes}},
     journal = {Canadian mathematical bulletin},
     pages = {523--534},
     year = {2005},
     volume = {48},
     number = {4},
     doi = {10.4153/CMB-2005-048-0},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2005-048-0/}
}
TY  - JOUR
AU  - Düvelmeyer, Nico
TI  - Angle Measures and Bisectors in Minkowski Planes
JO  - Canadian mathematical bulletin
PY  - 2005
SP  - 523
EP  - 534
VL  - 48
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2005-048-0/
DO  - 10.4153/CMB-2005-048-0
ID  - 10_4153_CMB_2005_048_0
ER  - 
%0 Journal Article
%A Düvelmeyer, Nico
%T Angle Measures and Bisectors in Minkowski Planes
%J Canadian mathematical bulletin
%D 2005
%P 523-534
%V 48
%N 4
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2005-048-0/
%R 10.4153/CMB-2005-048-0
%F 10_4153_CMB_2005_048_0

[1] [1] Averkov, G., On the geometry of simplices in Minkowski spaces. Stud. Univ. Žilina Math. Ser. 16(2003), 1–14. Google Scholar

[2] [2] Brass, P., Erdős distance problems in normed spaces. Comput. Geom. 6(1996), 195–214. Google Scholar

[3] [3] Busemann, H., Planes with analogues to Euclidean angular bisectors. Math. Scand. 36(1975), 5–11. Google Scholar

[4] [4] Düvelmeyer, N., On convex bodies all whose two-dimensional sections are equiframed. Submitted to Arch. Math. Google Scholar

[5] [5] Düvelmeyer, N., A new characterization of Radon curves via angular bisectors. J. Geom. 80(2004), no. 1–2, 75–81. Google Scholar

[6] [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] [7] Helfenstein, H., Problem 13. Canad. Math. Bull. 2(1959), 43. Google Scholar

[8] [8] Helfenstein, H., Solution to problem 13. Canad. Math. Bull. 4 (1961), 77–78. Google Scholar

[9] [9] Martini, H. and Swanepoel, K. J., Equiframed curves—a generalization of Radon curves. Monatsh. Math. 141(2004), 301–314. Google Scholar

[10] [10] Martini, H. and Swanepoel, K. J., The geometry of Minkowski spaces—a survey. II. Expo. Math. 22(2004), 92–144. Google Scholar

[11] [11] Pełczyński, A. and Szarek, S. J., On parallelepipeds of minimal volume containing a convex symmetric body in n . Math. Proc. Cambridge Philos. Soc. 109(1991), 125–148. Google Scholar

[12] [12] Radon, J., Über eine besondere Art ebener konvexer Kurven, Ber. Sächs. Akad.Wiss. Leipz. 68(1916), 131–134. Google Scholar

[13] [13] Thompson, A. C., Minkowski Geometry, Encyclopedia of Mathematics and its Applications 63, Cambridge University Press, Cambridge, 1996. Google Scholar

Cité par Sources :