Abstract

Gauges, or convex distance functions are, roughly speaking, norms without symmetry. In this paper we intend to quantify how asymmetric a planar gauge can be. We introduce asymmetry measures for smooth gauges and for strictly convex gauges, prove that they are invariant under isometries, and investigate lower and upper bounds for them. Identifying a gauge with a convex body containing the origin in its interior (the unit ball of the gauge), we also prove that all introduced asymmetry measures are continuous in the Hausdorff distance. Finally, we show that, modifying one of the constructed asymmetry measures, a certain duality principle holds.

Author information

Contact details are reproduced from the original publication and may be historical.

Vitor Balestro

Inst. de Matemática e Estatística, Universidade Federal Fluminense, 24210201 Niterói, Brazil

vitorbalestro@id.uff.br

Ralph Teixeira

Inst. de Matemática e Estatística, Universidade Federal Fluminense, 24210201 Niterói, Brazil

ralph@mat.uff.br

Keywords

  • Asymmetry measure
  • convex distance function
  • gauge space
  • Hausdorff distance
  • Mazur-Ulam theorem
  • orthogonality
  • symplectic form

Mathematics Subject Classification

52A10, 52A20, 52A21, 52A27, 52A38, 46B20

References

  1. J. Alonso, H. Martini, S. Wu: On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces, Aequationes Math. 83(1-2) (2012) 153–189.https://doi.org/10.1007/s00010-011-0092-zGoogle Scholar
  2. V. Balestro, H. Martini, R. Teixeira: Duality of gauges and symplectic forms in vector spaces, arXiv: 1901.03421 (2019), submitted.Google Scholar
  3. R. Brandenberg, S. König: Sharpening geometric inequalities using computable symmetry measures, Mathematika 61(3) (2015) 559–580.https://doi.org/10.1112/S0025579314000291Google Scholar
  4. R. Brandenberg, B. G. Merino: Minkowski concentricity and complete simplices, J. Math. Anal. Appl. 454(2) (2017) 981–994.https://doi.org/10.1016/j.jmaa.2017.05.031Google Scholar
  5. R. Brandenberg, B. G. Merino: The asymmetry of complete and constant width bodies in general normed spaces and the Jung constant, Israel J. Math. 218(1) (2017) 489–510.https://doi.org/10.1007/s11856-017-1471-5Google Scholar
  6. R. Brandenberg, B. G. Merino, T. Jahn, H. Martini: Is a complete, reduced set necessarily of constant width?, Adv. Geom. 19 (2019) 31–40.https://doi.org/10.1515/advgeom-2017-0058Google Scholar
  7. H. Busemann: The Geometry of Geodesics, Academic Press, New York (1955).Google Scholar
  8. Br. Grünbaum: Measures of symmetry for convex sets, in: Proc. Sympos. Pure Math., Vol. VII, V. Klee (ed.), American Mathematical Society, Providence (1963) 233–270.https://doi.org/10.1090/pspum/007/0156259Google Scholar
  9. P. C. Hammer: Diameters of convex bodies, Proc. Amer. Math. Soc. 5 (1954) 304–306.https://doi.org/10.1090/S0002-9939-1954-0061398-1Google Scholar
  10. T. Jahn: Extremal radii, diameter and minimum width in generalized Minkowski spaces, Rocky Mountain J. Math. 47(3) (2017) 825–848.https://doi.org/10.1216/RMJ-2017-47-3-825Google Scholar
  11. T. Jahn: Successive radii and ball operators in generalized Minkowski spaces, Adv. Geom. 17(3) (2017) 347–354.https://doi.org/10.1515/advgeom-2017-0012Google Scholar
  12. T. Jahn: Orthogonality in generalized Minkowski spaces, J. Convex Analysis 26(1) (2019) 49–76.https://doi.org/10.68381/jca26004Google Scholar
  13. B. G. Merino, T. Jahn, C. Richter: Uniqueness of circumcenters in generalized Minkowski spaces, J. Approx. Theory 237 (2019) 153–159.https://doi.org/10.1016/j.jat.2018.09.005Google Scholar
  14. A. Obst: A perimeter-based angle measure in generalized Minkowski planes, J. Geometry 110 (2019), no. 25.https://doi.org/10.1007/s00022-019-0480-4Google Scholar
  15. R. Schneider: Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications 151, Cambridge University Press, Cambridge (2014).Google Scholar
  16. V. Soltan: Affine diameters of convex bodies – a survey, Expo. Math. 23(1) (2005) 47–63.https://doi.org/10.1016/j.exmath.2005.01.019Google Scholar
  17. G. Tóth: Measures of Symmetry for Convex Sets and Stability, Universitext, Springer, Cham (2015).https://doi.org/10.1007/978-3-319-23733-6Google Scholar

Suggested citation

V. Balestro, H. Martini, R. Teixeira. “Asymmetry Measures for Convex Distance Functions.” Journal of Convex Analysis 27 (2020), No. 1, 117–138. https://doi.org/10.68381/jca27008

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2020. Rights now held by Banach Press.