Diameter of reduced spherical convex bodies

• Autorzy: Lassak M. , Musielak M.

Identyfikatory

DOI

10.1515/fascmath-2018-0020

• Języki publikacji: EN

Abstrakty

EN

The intersection L of two different non-opposite hemispheres of the unit sphere S² is called a lune. By Δ(L) we denote the distance of the centers of the semicircles bounding L. By the thickness Δ(C) of a convex body C ⊂ S² we mean the minimal value of Δ(L) over all lunes L ⊃ C. We call a convex body R ⊂ S² reduced provided Δ(Z) < Δ(R) for every convex body Z being a proper subset of R. Our aim is to estimate the diameter of R, where Δ(R) < π/2, in terms of its thickness.

Słowa kluczowe

EN

spherical convex body; spherical geometry; hemisphere; lune; width; constant width; thickness; diameter

• Wydawca: Wydawnictwo Politechniki Poznańskiej
• Czasopismo: Fasciculi Mathematici
• Rocznik: 2018
• Tom: Nr 61
• Strony: 103--108
• Opis fizyczny: Bibliogr. 14 poz.

Twórcy

autor

Lassak M.

• Instytut Matematyki i Fizyki University of Science and Technology 85-796 Bydgoszcz, Poland

autor

Musielak M.

• Instytut Matematyki i Fizyki University of Science and Technology 85-796 Bydgoszcz, Poland

Bibliografia

• [1] Besau F., Schuster F., Binary operations in spherical convex geometry, Indiana Univ. Math. J., 65(4)(2016), 1263-1288.
• [2] Danzer L., Grünbaum B., Klee V., Helly’s theorem and its relatives, Proc. of Symp. in Pure Math., vol. VII, Convexity, (1963), 99-180.
• [3] Ferreira O.P., Iusem A.N., Németh S.Z., Projections onto convex sets on the sphere, J. Global Optim., 57(2013), 663-676.
• [4] Gao F., Hug D., Schneider R., Intrinsic volumes and polar sets in spherical space, Math. Notae, 41(2003), 159-176.
• [5] Hadwiger H., Kleine Studie zur kombinatorischen Geometrie der Sphare, Nagoya Math. J., 8(1955), 45-48.
• [6] Han H., Nishimura T., Self-dual Wulff shapes and spherical convex bodies of constant width π/2, J. Math. Soc. Japan, 69(2017), 1475-1484.
• [7] Lassak М., Reduced convex bodies in the plane, Israel J. Math., 70(1990), 365-379.
• [8] Lassak М., Width of spherical convex bodies, Aequationes Math., 89(2015), 555-567.
• [9] Lassak М., Reduced spherical polygons, Colloq. Math., 138(2015), 205-216.
• [10] Lassak М., Martini H., Reduced convex bodies in finite-dimensional normed spaces - a survey, Results Math., 66(3-4)(2014), 405-426.
• [11] Lassak М., Musielak М., Reduced spherical convex bodies, Bull. Pol. Ac. Math., 66(2018), 87-97.
• [12] Lassak М., Musielak М., Spherical bodies of constant width, Aequationes Math., 92(2018), 627-640.
• [13] Musielak М., Covering a reduced spherical body by a disk, Ukr. Math. J., to appear (see also arXiv:1806.04246).
• [14] Van Brummelen G., Heavenly mathematics. The forgotten art of spherical trigonometry, Princeton University Press, Princeton, 2013.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).