Covering a reduced spherical body by a disk

M. Musielak

doi:10.37863/umzh.v72i10.6029

M. Musielak Univ. Sci. and Technology, Bydgoszcz, Poland

DOI: https://doi.org/10.37863/umzh.v72i10.6029

Keywords: spherical convex body, spherical geometry, hemisphere, lune, width, thickness, disk

Abstract

UDC 514

In this paper, the following theorems are proved: (1) every spherical convex body $W$ of constant width $\Delta (W) \geq \dfrac{\pi}{2}$ may be covered by a disk of radius $\Delta(W) + \arcsin \!\left(\dfrac{2\sqrt{3}}{3} \cos \dfrac{\Delta(W)}{2}\right) - \dfrac{\pi}{2};$ (2) every reduced spherical convex body $R$ of thickness $\Delta(R)<\dfrac{\pi}{2}$ may be covered by a disk of radius $\arctan \!\left(\sqrt{2} \tan \dfrac{\Delta(R)}{2}\right)\!.$

References

L. Danzer, B. Grunbaum, V. Klee, ¨ Helly’s theorem and its relatives, Proc. Sympos. Pure Math., Vol. VII, Convexity, p. 99 – 180, (1963). DOI: https://doi.org/10.1090/pspum/007/0157289

B. V. Dekster, The Jung theorem for spherical and hyperbolic spaces, Acta Math. Hungar., 67, № 4, 315–331 (1995), https://doi.org/10.1007/BF01874495 DOI: https://doi.org/10.1007/BF01874495

H. Han, T. Nishimura, Self-dual Wulff shapes and spherical convex bodies of constant width $pi/2$ , J. Math. Soc. Japan, 69, № 4, 1475 – 1484 (2017), https://doi.org/10.2969/jmsj/06941475 DOI: https://doi.org/10.2969/jmsj/06941475

M. Lassak, On the smallest disk containing a planar reduced convex body, Arch. Math., 80, no. 5, 553 – 560 (2003), https://doi.org/10.1007/s00013-003-4618-z DOI: https://doi.org/10.1007/s00013-003-4618-z

M. Lassak, Width of spherical convex bodies, Aequat. Math., 89, № 3, 555 – 567 (2015), https://doi.org/10.1007/s00010-013-0237-3 DOI: https://doi.org/10.1007/s00010-013-0237-3

M. Lassak, H. Martini, Reduced convex bodies in Euclidean space — a survey, Expo. Math., 29, no. 2, 204 – 219 (2011), https://doi.org/10.1016/j.exmath.2011.01.006 DOI: https://doi.org/10.1016/j.exmath.2011.01.006

M. Lassak, M. Musielak, Reduced spherical convex bodies, Bull. Pol. Acad. Sci. 66 , no. 1, 87--97 (2018), (see also arXiv:1607.00132v1)., https://doi.org/10.4064/ba8088-1-2018 DOI: https://doi.org/10.4064/ba8088-1-2018

M. Lassak, M. Musielak, Spherical bodies of constant width, Aequat. Math. 92, no. 4, 627--640, (2018), https://doi.org/10.1007/s00010-018-0558-3 DOI: https://doi.org/10.1007/s00010-018-0558-3

L. A. Masal’tsev, Incidence theorems in spaces of constant curvature, J. Math. Sci., 72, 3201 – 3206 (1994). DOI: https://doi.org/10.1007/BF01249519

J. Molnar, ´ Über eine Übertragung des Hellyschen Satzes in sphärische Räume. (German), Acta Math. Acad. Sci. Hung., 8, 315 – 318 (1957), https://doi.org/10.1007/BF02020320 DOI: https://doi.org/10.1007/BF02020320

D. A. Murray, Spherical trigonometry, Longmans Green and Co, London etc. (1900).

T. Nishimura, Y. Sakemi, Topological aspect of Wulff shapes, J. Math. Soc. Japan, 66, no. 1, 89 – 109 (2014), https://doi.org/10.2969/jmsj/06610089 DOI: https://doi.org/10.2969/jmsj/06610089