Diameter, width and thickness in the hyperbolic plane

In hyperbolic geometry there are several concepts to measure the breadth or width of a convex set. In the first part of the paper we collect them and compare their properties. Than we introduce a new concept to measure the width and thickness of a convex body. Correspondingly, we define three classe...

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Vydáno v:Journal of geometry Ročník 112; číslo 3
Hlavní autor: Horváth, Ákos G.
Médium: Journal Article
Jazyk:angličtina
Vydáno: Cham Springer International Publishing 01.12.2021
Springer Nature B.V
Témata:
Diameters
Geometry
Mathematics
Mathematics and Statistics
Shadows
Thickness
51M10
diameter of a convex set
Constant width property
hyperbolic plane
51M09
thickness of a convex set
width function
ISSN:0047-2468, 1420-8997
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Shrnutí:In hyperbolic geometry there are several concepts to measure the breadth or width of a convex set. In the first part of the paper we collect them and compare their properties. Than we introduce a new concept to measure the width and thickness of a convex body. Correspondingly, we define three classes of bodies, bodies of constant with, bodies of constant diameter and bodies having the constant shadow property, respectively. We prove that the property of constant diameter follows to the fulfilment of constant shadow property, and both of them are stronger as the property of constant width. In the last part of this paper, we introduce the thickness of a constant body and prove a variant of Blaschke’s theorem on the larger circle inscribed to a plane-convex body of given thickness and diameter.
Bibliografie:ObjectType-Article-1
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ISSN:0047-2468
1420-8997
DOI:10.1007/s00022-021-00613-3