Spherical bodies of constant width

Preprint English OPEN
Lassak, Marek ; Musielak, Michał (2018)
  • Subject: 52A55 | Mathematics - Metric Geometry

The intersection $L$ of two different non-opposite hemispheres $G$ and $H$ of a $d$-dimensional sphere $S^d$ is called a lune. By the thickness of $L$ we mean the distance of the centers of the $(d-1)$-dimensional hemispheres bounding $L$. For a hemisphere $G$ supporting a %spherical convex body $C \subset S^d$ we define ${\rm width}_G(C)$ as the thickness of the narrowest lune or lunes of the form $G \cap H$ containing $C$. If ${\rm width}_G(C) =w$ for every hemisphere $G$ supporting $C$, we say that $C$ is a body of constant width $w$. We present properties of these bodies. In particular, we prove that the diameter of any spherical body $C$ of constant width $w$ on $S^d$ is $w$, and that if $w < \frac{\pi}{2}$, then $C$ is strictly convex. Moreover, we are checking when spherical bodies of constant width and constant diameter coincide.
  • References (12)
    12 references, page 1 of 2

    [1] G. D. Chakerian, H. Groemer, Convex bodies of constant width, Convexity and its applications, 49-96, Birkhauser, Basel, 1983.

    [2] L. Danzer, B. Gru¨nbaum, V. Klee, Hellys theorem and its relatives, in Proc. of Symp. in Pure Math. vol. VII, Convexity, 1963, pp. 99-180.

    [3] B. Gonzalez Merino, T. Jahn, A. Polyanskii, G. Wachsmuth, Hunting for reduced polytopes, to appear in Discrete Comput. Geom. (see also arXiv:1701.08629v1).

    [4] H. Hadwiger, Kleine Studie zur kombinatorischen Geometrie der Spha¨re, Nagoya Math. J. 8 (1955), 45-48.

    [5] H. Han and T. Nishimura, Self-dual shapes and spherical convex bodies of constant width π/2, J. Math. Soc. Japan 69 (2017), 1475-1484.

    [6] M. Lassak, Width of spherical convex bodies, Aequationes Math. 89 (2015), 555-567.

    [7] M. Lassak, H. Martini, Reduced convex bodies in Euclidean space - a survey, Expositiones Math. 29 (2011), 204-219.

    [8] M. Lassak, M. Musielak, Reduced spherical convex bodies, to appear (see also arXiv:1607.00132v1).

    [9] K. Leichtweiss, Curves of constant width in the non-Euclidean geometry, Abh. Math. Sem. Univ. Hamburg 75 (2005), 257-284.

    [10] L. A. Santalo, Note on convex spherical curves, Bull. Amer. Math. Soc. 50 (1944), 528-534.

  • Metrics
    No metrics available
Share - Bookmark