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A perimeter-based angle measure in Minkowski planes

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Abstract

Measuring angles in the Euclidean plane is a well-known topic, but for general normed planes there exists a variety of different concepts. These can be of a special kind, e.g. also preserving special orthogonality types. But these concepts are no angle measures in the sense of measure theory since they are not additive. This motivates us to define a new angle measure for normed planes that is in fact a measure in the sense of measure theory. Furthermore, we look at related types of rotation and reflection.

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References

  1. Alonso, J., Benítez, C.: Orthogonality in normed linear spaces: a survey. I. Main properties. Extracta Math. 3(1), 1–15 (1988)

    MathSciNet  Google Scholar 

  2. Balestro, V., Horváth, A.G., Martini, H.: Angle measures, general rotations and roulettes in normed planes. Anal. Math. Phys 7, 549–575 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Balestro, V., Horváth, A.G., Martini, H., Teixeira, R.: Angles in normed spaces. Aequationes Math. 91(2), 201–236 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  4. Busemann, H.: Angular measure and integral curvature. Can. J. Math. 1, 279–296 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  5. Hilbert, D.: Grundlagen der Geometrie, 2nd edn. B. G. Teubner, Leipzig (1903)

    MATH  Google Scholar 

  6. Martini, H., Spirova, M.: Reflections in strictly convex Minkowski planes. Aequationes Math. 78(1–2), 71–85 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Martini, H., Spirova, M., Strambach, K.: Geometric algebra of strictly convex Minkowski planes. Aequationes Math. 88(1–2), 49–66 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Martini, H., Swanepoel, K.J.: The geometry of Minkowski spaces—a survey. II. Expo. Math. 22(2), 93–144 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Martini, H., Swanepoel, K.J., Weiß, G.: The geometry of Minkowski spaces—a survey. I. Expo. Math. 19(2), 97–142 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Thompson, A.C.: Minkowski Geometry, Volume 63 of Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (1996)

    Google Scholar 

  11. Thürey, V.: Angles and polar coordinates in real normed spaces. arxiv:0902.2731v2.pdf (2009)

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  1. Technische Universität Chemnitz, Chemnitz, Germany

    Martin Obst

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  1. Martin Obst
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Correspondence to Martin Obst.

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Obst, M. A perimeter-based angle measure in Minkowski planes. Aequat. Math. 92, 135–163 (2018). https://doi.org/10.1007/s00010-017-0526-3

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  • DOI: https://doi.org/10.1007/s00010-017-0526-3

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