Skip to main content
Log in

The asymmetry of complete and constant width bodies in general normed spaces and the Jung constant

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper we state a one-to-one connection between the maximal ratio of the circumradius and the diameter of a body (the Jung constant) in an arbitrary Minkowski space and the maximal Minkowski asymmetry of the complete bodies within that space. This allows to generalize and unify recent results on complete bodies and to derive a necessary condition on the unit ball of the space, assuming a given body to be complete. Finally, we state several corollaries, e.g. concerning the Helly dimension or the Banach–Mazur distance.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. Alexander, The width and diameter of a simplex, Geometriae Dedicata 6, (1977) 87–94.

    Article  MathSciNet  MATH  Google Scholar 

  2. I. Bárány, M. Katchalski and J. Pach, Quantitative Helly-type theorems, Proc. Amer. Math. Soc. 86, (1982) 109–114.

    Article  MathSciNet  MATH  Google Scholar 

  3. I. Bárány, M. Katchalski and J. Pach, Helly’s theorem with volumes, Amer. Math. Monthly 91, (1984) 362–365.

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Belloni and R. M. Freund, On the symmetry function of a convex set, Math. Program. 111, (2008) 57–93.

    Article  MathSciNet  MATH  Google Scholar 

  5. U. Betke and M. Henk, Estimating sizes of a convex body by successive diameters and widths, Mathematika 39, (1992) 247–257.

    Article  MathSciNet  MATH  Google Scholar 

  6. U. Betke and M. Henk, A generalization of Steinhagen’s theorem, Abh. Math. Sem. Univ. Hamburg 63, (1993) 165–176.

    Article  MathSciNet  MATH  Google Scholar 

  7. F. Bohnenblust, Convex regions and projections in Minkowski spaces, Ann. ofMath. 39 (2), (1938) 301–308.

    Article  MathSciNet  MATH  Google Scholar 

  8. V. Boltyanski and H. Martini, Jung’s theorem for a pair of Minkowski spaces, Adv. Geom. 6, (2006) 645–650.

    Article  MathSciNet  MATH  Google Scholar 

  9. T. Bonnesen and W. Fenchel, Theory of convex bodies, BCS Associates, Moscow, ID, 1987, Translated from the German and edited by L. Boron, C. Christenson and B. Smith.

  10. K. Böröczky, M. A. Hernández Cifre and G. Salinas, Optimizing area and perimeter of convex sets for fixed circumradius and inradius, Monatsh. Math. 138, (2003) 95–110.

    Article  MathSciNet  MATH  Google Scholar 

  11. R. Brandenberg, Radii of convex bodies, Ph.D. thesis, Zentrum Mathematik, Technische Universität München, 2002.

    Google Scholar 

  12. R. Brandenberg, Radii of regular polytopes, Discrete Comput. Geom. 33, (2005) 43–55.

    Article  MathSciNet  MATH  Google Scholar 

  13. R. Brandenberg and S. König, No dimension-independent core-sets for containment under homothetics, Discrete Comput. Geom. 49, (2013) 3–21.

    Article  MathSciNet  MATH  Google Scholar 

  14. R. Brandenberg and S. König, Sharpening geometric inequalities using computable symmetry measures, Mathematika 61, (2015) 559–580.

    Article  MathSciNet  MATH  Google Scholar 

  15. L. Caspani and P. L. Papini, On constant width sets in Hilbert spaces and around, J. Convex Anal. 22, (2015) 889–900.

    MathSciNet  MATH  Google Scholar 

  16. L. Danzer, B. Grünbaum and V. Klee, Helly’s theorem and its relatives, in Proc. Sympos. Pure Math., Vol. VII, Amer. Math. Soc., Providence, R.I., 1963, pp. 101–180.

    Google Scholar 

  17. V. L. Dolnikov, The Jung constant in ln 1, Mat. Zametki 42, (1987) 519–526, 622.

    MathSciNet  Google Scholar 

  18. H. G. Eggleston, Measure of asymmetry of convex curves of constant width and restricted radii of curvature, Quart. J. Math., Oxford Ser. 3 (2) (1952), 63–72.

    Article  MathSciNet  MATH  Google Scholar 

  19. H. G. Eggleston, Sets of constant width, in Convexity, Cambridge Tracts in Mathematics and Mathematical Physics, No. 47, Cambridge University Press, New York, 1958, pp. 122–135.

    Google Scholar 

  20. H. G. Eggleston, Sets of constant width in finite dimensional Banach spaces, Israel J. Math. 3, (1965) 163–172.

    Article  MathSciNet  MATH  Google Scholar 

  21. B. González Merino, On the ratio between successive radii of a symmetric convex body, Math. Inequal. Appl. 16, (2013) 569–576.

    MathSciNet  MATH  Google Scholar 

  22. P. Gritzmann and V. Klee, Inner and outer j-radii of convex bodies in finite-dimensional normed spaces, Discrete Comput. Geom. 7, (1992) 255–280.

    Article  MathSciNet  MATH  Google Scholar 

  23. H. Groemer, On complete convex bodies, Geom. Dedicata 20, (1986) 319–334.

    Article  MathSciNet  MATH  Google Scholar 

  24. B. Grünbaum, Measures of symmetry for convex sets, in Proc. Sympos. Pure Math., Vol. VII, Amer. Math. Soc., Providence, R.I., 1963, pp. 233–270.

    Google Scholar 

  25. Q. Guo and S. Kaijser, Approximation of convex bodies by convex bodies, Northeast. Math. J. 19, (2003) 323–332.

    MathSciNet  MATH  Google Scholar 

  26. M. Henk, A generalization of Jung’s theorem, Geom. Dedicata 42, (1992) 235–240.

    Article  MathSciNet  MATH  Google Scholar 

  27. M. Henk and M. A. Hernández Cifre, Intrinsic volumes and successive radii, J. Math. Anal. Appl. 343, (2008) 733–742.

    Article  MathSciNet  MATH  Google Scholar 

  28. H. Jin, Asymmetry for convex body of revolution, Wuhan Univ. J. Nat. Sci. 20, (2015) 97–100.

    Article  MathSciNet  MATH  Google Scholar 

  29. H. Jin and Q. Guo, Asymmetry of convex bodies of constant width, Discrete Comput. Geom. 47, (2012) 415–423.

    Article  MathSciNet  MATH  Google Scholar 

  30. H. Jin and Q. Guo, A note on the extremal bodies of constant width for the Minkowski measure, Geom. Dedicata 164, (2013) 227–229.

    Article  MathSciNet  MATH  Google Scholar 

  31. H. Jung, Über die kleinste Kugel, die eine räumliche Figur einschliesst, J. Reine Angew. Math. 123, (1901) 241–257.

    MathSciNet  MATH  Google Scholar 

  32. K. Leichtweiss, Zwei Extremalprobleme der Minkowski-Geometrie, Math. Z. 62, (1955) 37–49.

    Article  MathSciNet  MATH  Google Scholar 

  33. H. Martini and S. Wu, Complete sets need not be reduced in Minkowski spaces, Beitr. Algebra Geom. 56, (2015) 533–539.

    Article  MathSciNet  MATH  Google Scholar 

  34. J. P. Moreno, P. L. Papini and R. R. Phelps, Diametrically maximal and constant width sets in Banach spaces, Canad. J. Math. 58, (2006) 820–842.

    Article  MathSciNet  MATH  Google Scholar 

  35. J. P. Moreno and R. Schneider, Diametrically complete sets in Minkowski spaces, Israel J. Math. 191, (2012) 701–720.

    Article  MathSciNet  MATH  Google Scholar 

  36. J. P. Moreno and R. Schneider, Structure of the space of diametrically complete sets in a Minkowski space, Discrete Comput. Geom. 48, (2012) 467–486.

    Article  MathSciNet  MATH  Google Scholar 

  37. G. Y. Perelman, On the k-radii of a convex body, Sibirsk. Mat. Zh. 28, (1987) 185–186.

    MathSciNet  Google Scholar 

  38. G. T. Sallee, Sets of constant width, the spherical intersection property and circumscribed balls, Bull. Austral. Math. Soc. 33, (1986) 369–371.

    Article  MathSciNet  MATH  Google Scholar 

  39. L. Santaló, Sobre los sistemas completos de desigualdades entre tres elementos de una figura convexa planas, Math. Notae 17, (1961) 82–104.

    MathSciNet  Google Scholar 

  40. R. Schneider, Stability for some extremal properties of the simplex, J. Geom. 96, (2009) 135–148.

    Article  MathSciNet  MATH  Google Scholar 

  41. R. Schneider, Convex bodies: the Brunn-Minkowski theory, expanded ed., Encyclopedia of Mathematics and its Applications, Vol. 151, Cambridge University Press, Cambridge, 2014.

    Google Scholar 

  42. P. R. Scott, Sets of constant width and inequalities, Quart. J. Math. Oxford Ser. 32 (2), (1981) 345–348.

    Article  MathSciNet  MATH  Google Scholar 

  43. V. P. Soltan, A theorem on full sets, Dokl. Akad. Nauk SSSR 234, (1977) 320–322.

    MathSciNet  Google Scholar 

  44. P. Steinhagen, Über die größte Kugel in einer konvexen Punktmenge, Abh. Math. Sem. Univ. Hamburg 1, (1922) 15–26.

    Article  MathSciNet  MATH  Google Scholar 

  45. B. Sz.-Nagy, Ein Satz über Parallelverschiebungen konvexer Körper, Acta Sci. Math. Szeged 15, (1954) 169–177.

    MathSciNet  MATH  Google Scholar 

  46. S. Vrécica, A note on sets of constant width, Publ. Inst. Math. (Beograd) (N.S.) 29(43), (1981) 289–291.

    MathSciNet  MATH  Google Scholar 

  47. D. Yost, Irreducible convex sets, Mathematika 38, (1991) 134–155.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to René Brandenberg.

Additional information

The second author was partially supported by MINECO-FEDER project reference MTM2012-34037, Spain, and by Consejería de Industria, Turismo, Empresa e Innovación de la CARM through Fundación Séneca, Agencia de Ciencia y Tecnolog ía de la Región de Murcia, Programa de Formación Postdoctoral de Personal Investigador, project reference 19769/PD/15.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Brandenberg, R., Merino, B.G. The asymmetry of complete and constant width bodies in general normed spaces and the Jung constant. Isr. J. Math. 218, 489–510 (2017). https://doi.org/10.1007/s11856-017-1471-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-017-1471-5

Navigation