Spherical convexity and hyperbolic metric

Toshiyuki Sugawa

Spherical convexity and hyperbolic metric

INF - Computer and Mathematical Sciences

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

Abstract

Let Ω be a domain in C with hyperbolic metric λ_Ω(z)|dz| with Gaussian curvature −4. Mejía and Minda proved in their 1990 paper that Ω is (Euclidean) convex if and only if d(z, ∂Ω)λ_Ω(z) ≥ 1/2 for z ∈ Ω, where d(z, ∂Ω) denotes the Euclidean distance from z to the boundary ∂Ω. In the present note, we will provide similar characterizations of spherically convex domains in terms of the spherical density of the hyperbolic metric.

2010 Mathematics Subject Classification. Primary 30F45; Secondary 30C80, 51M10.

MSC Codes Primary 30F45, Secondary 30C80, 51M10

Original language	English
Journal	Unknown Journal
Publication status	Published - 2017 Apr 25

Keywords

. hyperbolic metric
Spherically convex
Tree

ASJC Scopus subject areas

General

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N2 - Let Ω be a domain in C with hyperbolic metric λΩ(z)|dz| with Gaussian curvature −4. Mejía and Minda proved in their 1990 paper that Ω is (Euclidean) convex if and only if d(z, ∂Ω)λΩ(z) ≥ 1/2 for z ∈ Ω, where d(z, ∂Ω) denotes the Euclidean distance from z to the boundary ∂Ω. In the present note, we will provide similar characterizations of spherically convex domains in terms of the spherical density of the hyperbolic metric.2010 Mathematics Subject Classification. Primary 30F45; Secondary 30C80, 51M10.MSC Codes Primary 30F45, Secondary 30C80, 51M10

AB - Let Ω be a domain in C with hyperbolic metric λΩ(z)|dz| with Gaussian curvature −4. Mejía and Minda proved in their 1990 paper that Ω is (Euclidean) convex if and only if d(z, ∂Ω)λΩ(z) ≥ 1/2 for z ∈ Ω, where d(z, ∂Ω) denotes the Euclidean distance from z to the boundary ∂Ω. In the present note, we will provide similar characterizations of spherically convex domains in terms of the spherical density of the hyperbolic metric.2010 Mathematics Subject Classification. Primary 30F45; Secondary 30C80, 51M10.MSC Codes Primary 30F45, Secondary 30C80, 51M10

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