Completeness and constant width in spherical and hyperbolic spaces
type: Item
instance of: scholarly article
title: Completeness and constant width in spherical and hyperbolic spaces
zbMATH Open document ID: 0867.52001
DOI: 10.1007/BF01874493
published in: Acta Mathematica Hungarica
publication date: 1997-07-10T00:00:00Z
review text: A compact set in a Euclidean (\(E^n\)), spherical (\(S^n\)), or hyperbolic (\(H^n\)) space is called complete if one cannot add a point without increasing its diameter. Following the study of complete sets initiated by \textit{E. Meissner} [Naturforsch. Ges. Zürich 56, 42-50 (1911; JFM 42.0091.01)]) in a Euclidean setting, the author shows examples of complete sets on the sphere and proves that the complete set are the bodies of constant width in \(E^n\) or \(H^n\). Regarding \(S^n\), this characterization is only true for convex sets. Moreover, if the complete convex set on \(S^2\) has diameter \(D>\pi/2\), then it is smooth. The author shows examples of complete non-convex sets on \(S^2\) that do not have constant width.
Mathematics Subject Classification ID: 52A05
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 868510
zbMATH Keywords: Euclidean space
zbMATH Keywords: spherical space
zbMATH Keywords: hyperbolic space
zbMATH Keywords: completeness
zbMATH Keywords: JFM 42.0091.01
zbMATH Keywords: constant width
author: B. V. Dekster
MaRDI profile type: Publication
cites work: Q3764803
cites work: In- and circumcenters of manifolds of constant width
cites work: Large spherical simplexes
full work available at URL: https://doi.org/10.1007/bf01874493
OpenAlex ID: W2020451100
description: scientific article; zbMATH DE number 868510