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Perimeter, Diameter and Area of Convex Sets in the Hyperbolic Plane
Author(s) -
Gallego Eduardo,
Solanes Gil
Publication year - 2001
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s002461070100223x
Subject(s) - mathematics , regular polygon , limit (mathematics) , perimeter , plane (geometry) , combinatorics , convex body , euclidean geometry , constant (computer programming) , convex hull , mathematical analysis , hyperbolic geometry , sequence (biology) , upper and lower bounds , convex set , geometry , convex optimization , differential geometry , chemistry , computer science , programming language , biochemistry
The paper studies the relation between the asymptotic values of the ratios area/length ( F /L) and diameter/length ( D / L ) of a sequence of convex sets expanding over the whole hyperbolic plane. It is known that F / L goes to a value between 0 and 1 depending on the shape of the contour. In the paper, it is first of all seen that D / L has limit value between 0 and 1/2 in strong contrast with the euclidean situation in which the lower bound is 1/π ( D / L = 1/π if and only if the convex set has constant width). Moreover, it is shown that, as the limit of D / L approaches 1/2, the possible limit values of F / L reduce. Examples of all possible limits F / L and D / L are given.