Abstract
This chapter concerns the geometry of convex bodies on the d-dimensional sphere S d. We concentrate on the results based on the notion of width of a convex body C ⊂ S d determined by a supporting hemisphere of C. Important tools are the lunes containing C. The supporting hemispheres take over the role of the supporting half-spaces of a convex body in Euclidean space, and lunes the role of strips. Also essential is the notion of thickness of C, i.e., its minimum width. In particular, we describe properties of reduced spherical convex bodies and spherical bodies of constant width. The last notion coincides with the notions of complete bodies and bodies of constant diameter on S d. The results reminded and commented on here concern mostly the width, thickness, diameter, perimeter, area and extreme points of spherical convex bodies, reduced bodies and bodies of constant width.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
P.V. Araújo, Barbier’s theorem for the sphere and the hyperbolic plane. L’Enseign. Math. 42, 295–309 (1966)
M.J.C. Baker, A spherical Helly-type theorem. Pac. J. Math. 23, 1–3 (1967)
E. Barbier, Note sur le problème de l’aiguille et le jeu du joint couvert. J. Math. Pures Appl. 5, 273–286 (1860)
R.V. Benson, Euclidean Geometry and Convexity (McGraw-Hill Book Company, New York, 1966)
F. Besau, T. Hack, P. Pivovarov, F.E. Schuster, Spherical Centroid Bodies (2019). ar**v:1902.10614
F. Besau, S. Schuster, Binary operations in spherical convex geometry. Indiana Univ. Math. J. 65(4), 1263–1288 (2016)
K. Bezdek, Illuminating spindle convex bodies and minimizing the volume of spherical sets of constant width. Discrete Comput. Geom. 47(2), 275–287 (2012)
K. Bezdek, A new look at the Blaschke-Leichtweiss theorem. ar**v:101.00538v1
K. Bezdek, Z. Lángi, From spherical to Euclidean illumination. Monatsh. Math. 192(3), 483–492 (2020)
W. Blaschke, Einige Bemerkungen über Kurven and Flaschen von konstanter Breite. Ber. Verh. Sächs. Akad. Leipzig 67, 290–297 (1915)
W. Blaschke, Konvexe Bereiche gegebener konstanter Breite und kleinsten Inthalts. Math. Ann. 76, 504–513 (1915)
T. Bonnesen, T.W. Fenchel, Theorie der konvexen Körper. Springer, Berlin (1934) (English translation: Theory of Convex Bodies, BCS Associates, Moscow, Idaho, 1987)
K.J. Böroczky, A. Sagemeister, The isodiametric problem on the sphere and the hyperbolic space. Acta Math. Hungar. 160(1), 13–32 (2020)
G.D. Chakerian, H. Groemer, Convex bodies of constant width, in The Collection of Surveys “Convexity and Its Applications, ed. by P.M. Gruber, J.M. Wills (Birkhäuser, Basel 1983), pp. 49–96
C.Y. Chang, C. Liu, Z. Su, The perimeter and area of reduced spherical polygons of thickness π∕2. Results Math. 75(4), 135 (2020)
L. Danzer, B. Grünbaum, V. Klee, Helly’s theorem and its relatives, in Proc. of Symp. in Pure Math., vol. VII, ed. by V. Klee (Convexity, 1963), pp. 99–180
B.V. Dekster, Completness and constant width in spherical and hyperbolic spaces. Acta Math. Hungar. 67, 289–300 (1995)
B.V. Dekster, The Jung theorem for spherical and hyperbolic space. Acta Math. Hungar. 67(4), 315–331 (1995)
B.V. Dekster, Double normals characterize bodies of constant width in Riemannian manifolds, in Geometric Analysis and Nonlinear Partial Differential Equations. Lecture Notes in Pure an Applied Mathematics, vol. 144 (New York, 1993), pp. 187–201
B.V. Dekster, B. Wegner, Constant width and transformality in spheres. J. Geom. 56(1–2), 25–33 (1996)
H.G. Eggleston, Convexity (Cambridge University Press, Cambridge, 1958)
L. Euler, it De curvis triangularibus (on triangular curves). Acta Akad. Sci. Imper. Petropolitinae 1778(II), 3–30 (1781); Opera Omnia, Series, Vol. 28, pp. 298–321
E. Fabińska, M. Lassak, Reduced bodies in normed planes. Israel J. Math. 161, 75–87 (2007)
O.P. Ferreira, A.N. Iusem, S.Z. Németh, Projections onto convex sets on the sphere. J. Global Optim. 57, 663–676 (2013)
B. Fuchssteiner, W. Lucky, Convex Cones. North-Holland Mathematics Studies, vol. 56 (North-Holland, Amsterdam, 1981)
E. Gallego, A. Reventós, G. Solanes, E. Teufel, Width of convex bodies in spaces of convex curvature. Manuscr. Math. 126(1), 115–134 (2008)
F. Gao, D. Hug., R. Schneider, Intrinsic volumes and polar sets in spherical space. Math. Notae 41, 159–176 (2003)
B. Gonzalez Merino, T. Jahn, A. Polyanskii, G. Wachsmuth, Hunting for reduced polytopes. Discrete Comput. Geom. 60(3), 801–808 (2018)
H. Groemer, Extremal convex sets. Monatsh. Math. 96, 29–39 (1983)
Q. Guo, Convexity Theory on Spherical Spaces (Scientia Sinica Mathematica, 2020)
Q. Guo, Y. Peng, Spherically convex sets and spherically convex functions. J. Convex Anal. 28, 103–122 (2021)
H. Hadwiger, Kleine Studie zur kombinatorischen Geometrie der Sphäre. Nagoya Math. J. 8, 45–48 (1955)
H. Hadwiger, Ausgewählte Probleme der kombinatorischen Geometrie des Euklidischen und Sphärischen Raumes. L’Enseign. Math. 3, 73–75 (1957)
N.N. Hai, P.T. An, A Generalization of Blaschke’s convergence theorem in metric spaces. J. Conv. Anal. 4, 1013–1024 (2013)
H. Han, Maximum and minimum of support functions. ar**v:1701.08956v2
H. Han, T. Nishimura, Self-dual shapes and spherical convex bodies of constant width π∕2. J. Math. Soc. Jpn. 69, 1475–1484 (2017)
H. Han, T. Nishimura, Spherical method for studying Wulff shapes and related topics, in The volume “Singularities in Generic Geometry” ed. by Math. Soc. Japan. Adv. Stud. in Pure Math., vol. 78 (2018), pp. 1–53
H. Han, T. Nishimura, Wulff shapes and their duals—RIMS, Kyoto University. http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2049-04.pdf~
H. Han, D. Wu, Constant diameter and constant width of spherical convex bodies. Aequat. Math. 95, 167–174 (2021)
E. Heil, Kleinste konvexe Körper gegebener Dicke, Preprint No. 453, Fachbereich Mathematik der TH Darmstadt (1978)
M.A. Hernandez Cifre, A.R. Martinez Fernández, The isodiametric problem and other inequalities in the constant curvature 2-spaces. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 109(2), 315–325 (2015)
A.G. Horváth, Diameter, width and thickness in hyperbolic plane. ar**v:2011.14739v1
B. Jessen, Über konvexe Punktmengen konstanter Breite. Math. Z. 29(1), 378–380 (1929)
T. Kubota, On the maximum area of the closed curve of a given perimeter. Tokyo Matk. Ges. 5, 109–119 (1909)
M. Lassak, Reduced convex bodies in the plane. Israel J. Math. 70(3), 365–379 (1990)
M. Lassak, On the smallest disk containing a planar reduced convex body. Arch. Math. 80, 553–560 (2003)
M. Lassak, Area of reduced polygons. Publ. Math. 67, 349–354 (2005)
M. Lassak, Approximation of bodies of constant width and reduced bodies in a normed plane. J. Convex Anal. 19(3). 865–874 (2012)
M. Lassak, Width of spherical convex bodies. Aequat. Math. 89, 555–567 (2015)
M. Lassak, Reduced spherical polygons. Colloq. Math. 138, 205–216 (2015)
M. Lassak, Diameter, width and thickness of spherical reduced convex bodies with an application to Wulff shapes. Beitr Algebra Geom. 61, 369–379 (2020) (earlier in ar**v:193.041148)
M. Lassak, When is a spherical convex body of constant diameter a body of constant width? Aequat. Math. 94, 393–400 (2020)
M. Lassak, Complete spherical convex bodies. J. Geom. 111(2), 35 (2020), 6p (a corrected version is in ar**v:2004.1011v.2)
M. Lassak, Approximation of reduced spherical convex bodies. J. Convex Anal. (to appear). ar**v:2106.00118v1
M. Lassak, H. Martini, Reduced convex bodies in Euclidean space—a survey. Expositiones Math. 29, 204–219 (2011)
M. Lassak, H. Martini, Reduced convex bodies in finite-dimensional normed spaces—a survey. Results Math. 66(3–4), 405–426 (2014)
M. Lassak M., M. Musielak, Reduced spherical convex bodies. Bull. Pol. Ac. Math. 66, 87–97 (2018)
M. Lassak, M. Musielak, Spherical bodies of constant width. Aequat. Math. 92, 627–640 (2018)
M. Lassak, M. Musielak, Diameter of reduced spherical convex bodies. Fasciculi Math. 61, 83–88 (2018)
S.B. Lay, Convex Sets and Its Applications. A Willey-Interscience Publication (Wiley, New York, 1982)
H. Lebesgue, Sur quelques questions de minimum, relatives aux courbes orbiformes, et sur leurs rapports avec le calcul des variations. J. Math. Pures Appl. 8(4), 67–96 (1921)
K. Leichtweiss, Curves of constant width in the non-Euclidean geometry. Abh. Math. Sem. Univ. Hamburg 75, 257–284 (2005)
C. Liu, Y. Chang, Z. Su, The area of reduced spherical polygons. ar**v:2009.13268v1
H. Maehara, H. Martini, Geometric probability on the sphere. Jahresber. Dtsch. Math.-Ver. 119(2), 93–132 (2017)
M. Martini, L. Montejano, D. Oliveros, Bodies of Constant Width. An Introduction to Convex Geometry with Applications (Springer Nature Switzerland AG, 2019)
H. Martini, K.J. Swanepoel, Non-planar simplices are not reduced. Publ. Math. Debrecen 64, 101–106 (2004)
E. Meissner, Über Punktmengen konstanter Breite, Vjschr. Naturforsch. Ges. Zürich 56, 42–50 (1911)
J. Molnár, Über eine Übertragung des Hellyschen Satzes in sphärischen Räume. Acta Mat. Hungar. 8, 315–318 (1957)
D.A. Murray, Spherical Trigonometry (Longmans Green, London, Bombay and Calcutta, 1900)
M. Musielak, Covering a reduced spherical body by a disk. Ukr. Math. J. 72(10), 1400–1409 (2020)
J. Pál, Über ein elementares Variationsproblem. Bull. de l’Acad. de Dan. 3(2), 35 (1920)
A. Papadopoulos, Metric Spaces, Convexity and Nonpositive Curvature, 2nd edn. IRMA Lectures in Mathematics and Theoretical Physics. European Mathematical Society (Zürich, 2014)
A. Papadopoulos, On the works of Euler and his followers on spherical geometry. Ganita Bharati 38(1), 53–108 (2014)
A. Papadopoulos, Three theorem of Menelaus. Am. Math. Monthly 126(7), 610–619 (2019)
A. Pimpinelli, J. Vilain, Physics of Crystal Growth. Monographs and Texts in Statistical Physics (Cambridge University Press, Cambridge, New York, 1998)
K. Reidemeister, Über Körper konstanten Durchmessers. Math. Z. 10, 214–216 (1921)
B.A. Rosenfeld, A History of Non-Euclidean Geometry (Springer, New York, 2012)
L.A. Santaló, Note on convex spherical curves. Bull. Am. Math. Soc. 50, 528–534 (1944)
L.A. Santaló, Properties of convex figures on a sphere. Math. Notae 4, 11–40 (1944)
L.A. Santaló, Convex regions on the n-dimensional spherical surface. Ann. Math. 47, 448–459 (1946)
R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, second expanded edition, Encyclopedia of Mathematics and its Applications, vol. 151 (Cambridge University Press, 2013, Cambridge, 2014)
Y. Shao, Q. Guo, An analytic approach to spherically convex sets in S n−1. J. Math. (PRC) 38(3), 473–480 (2018)
I. Todhunter, Spherical Trigonometry, 5th edn. (Macmillan, London, 1886)
G. Toth, Measures of Symmetry for Convex Sets and Stability (Springer, Cham, 2015)
G. Van Brummelen, Heavenly Mathematics. The Forgotten Art of Spherical Trigonometry (Princeton University Press, Princeton, 2013)
M.A. Whittlesey, Spherical Geometry and Its Applications (CRC Press, Taylor and Francis Group, 2020)
G. Wulff, Zur Frage der Geschwindigkeit des Wachstrums und der Auflösung der Krystallflaäschen. Z. Kryst. Mineral. 34, 449–530 (1901)
I.M. Yaglom, V.G. Boltyanskij, Convex Figures (Moscow 1951). (English translation: Holt, Rinehart and Winston, New York 1961)
C. Zălinescu, On the separation theorems for convex sets on the unit sphere. ar**v:2103.04321v1
X. Zhou, Q. Guo, Compositions and valuations on spherically convex sets. Wuhan Univ. J. Nat. Sci. (2020)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Lassak, M. (2022). Spherical Geometry—A Survey on Width and Thickness of Convex Bodies. In: Papadopoulos, A. (eds) Surveys in Geometry I. Springer, Cham. https://doi.org/10.1007/978-3-030-86695-2_2
Download citation
DOI: https://doi.org/10.1007/978-3-030-86695-2_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-86694-5
Online ISBN: 978-3-030-86695-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)