Asymptotic Behaviour of λ-Convex Sets in the Hyperbolic Plane

Eduardo Gallego; Agustí Reventós

doi:10.1023/A:1005130211872

Asymptotic Behaviour of λ-Convex Sets in the Hyperbolic Plane

Eduardo Gallego, Agustí Reventós

Department of Mathematics

Research output: Contribution to journal › Article › Research › peer-review

12 Citations (Scopus)

Abstract

It is known that the limit Area/Length for a sequence of convex sets expanding over the whole hyperbolic plane is less than or equal to 1, and exactly 1 when the sets considered are convex with respect to horocycles. We consider geodesics and horocycles as particular cases of curves of constant geodesic curvature λ with 0 ≤ λ ≤ 1 and we study the above limit Area/Length as a function of the parameter λ.

Original language	English
Pages (from-to)	275-289
Journal	Geometriae Dedicata
Volume	76
Issue number	3
DOIs	https://doi.org/10.1023/A:1005130211872
Publication status	Published - 1 Jan 1999

Keywords

Hyperbolic plane
Integral geometry
λ-convex set

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abstract = "It is known that the limit Area/Length for a sequence of convex sets expanding over the whole hyperbolic plane is less than or equal to 1, and exactly 1 when the sets considered are convex with respect to horocycles. We consider geodesics and horocycles as particular cases of curves of constant geodesic curvature λ with 0 ≤ λ ≤ 1 and we study the above limit Area/Length as a function of the parameter λ.",

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AU - Reventós, Agustí

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N2 - It is known that the limit Area/Length for a sequence of convex sets expanding over the whole hyperbolic plane is less than or equal to 1, and exactly 1 when the sets considered are convex with respect to horocycles. We consider geodesics and horocycles as particular cases of curves of constant geodesic curvature λ with 0 ≤ λ ≤ 1 and we study the above limit Area/Length as a function of the parameter λ.

AB - It is known that the limit Area/Length for a sequence of convex sets expanding over the whole hyperbolic plane is less than or equal to 1, and exactly 1 when the sets considered are convex with respect to horocycles. We consider geodesics and horocycles as particular cases of curves of constant geodesic curvature λ with 0 ≤ λ ≤ 1 and we study the above limit Area/Length as a function of the parameter λ.

KW - Hyperbolic plane

KW - Integral geometry

KW - λ-convex set

U2 - 10.1023/A:1005130211872

DO - 10.1023/A:1005130211872

M3 - Article

VL - 76

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EP - 289

JO - Geometriae Dedicata

JF - Geometriae Dedicata

SN - 0046-5755

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