Asymptotic behavior of convex sets in the hyperbolic plane
type: Item
instance of: scholarly article
title: Asymptotic behavior of convex sets in the hyperbolic plane
zbMATH Open document ID: 0592.53051
published in: Journal of Differential Geometry
publication date: 1985-01-01T00:00:00Z
review text: We solve in the negative the following conjecture, posed by \textit{L. A. Santaló} and \textit{I. Yañez} in 1972 [J. Appl. Probab. 9, 140-157 (1972; Zbl 0231.60010)]: Let C(t) be a family of bounded closed sets in the hyperbolic plane, depending on the parameter t (0\(\leq t)\) and such that \(C(t_ 1)\subset C(t_ 2)\) for \(t_ 1<t_ 2\). Assume that for any point P of the plane there is a value \(t_ p\) of t such that, for all \(t\geq t_ p\), we have \(P\in C(t)\). If F(t) and L(t) denote respectively the area and length of C(t), prove that \(\lim_{t\to \infty}L(t)/F(t)=(-K)^{1/2}\) where \(K<0\) is the curvature of the hyperbolic plane. In fact we construct for each \(\lambda \in [(-K)^{1/2},\infty]\) a sequence of polygonal convex sets C(t) expanding over the whole hyperbolic plane with \(\lim_{t\to \infty}L(t)/F(t)=\lambda\). We also prove that the above sequence can be modified to obtain counterexamples with smooth boundary.
Mathematics Subject Classification ID: 53C65
Mathematics Subject Classification ID: 52A22
zbMATH DE Number: 3951499
zbMATH Keywords: area
zbMATH Keywords: length
zbMATH Keywords: hyperbolic plane
zbMATH Keywords: polygonal convex sets
author: Eduardo Gallego
author: Agustí Reventós
Wikidata QID: Q115187346
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214439464
OpenAlex ID: W2997290782
DOI: 10.4310/JDG/1214439464
description: scientific article; zbMATH DE number 3951499