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Convexity in hyperbolic spaces
by
Eduardo Gallego
Universitat Autónoma de Barcelona, Spain
Coauthors: A.A. Borisenko, A. Reventós, G. Solanes
Oral Communication
Hyperbolic plane
In the study of some problems in geometric probability appears the limit of the quotient between the area F of a convex set and the length L of its boundary. For instance, if \sigma is the random variable length of a chord of a given convex set, the expected value of \sigma is given by E(\sigma)=\piF/L. In the euclidean plane it is clear that this quotient, when the convex set becomes ``very large", tends to \infty. But, as it was pointed out by L.A. Santaló and I. Yañez in 1972, this is no longer true in the hyperbolic plane. In fact they proved that for a certain class of convex sets in the hyperbolic plane, the horocyclic convex sets, the limit F/L is 1. Latter it was shown that this limit can attain, in the hyperbolic plane, any value between 0 and 1.
Since horocycles are curves of geodesic curvature +/- 1 and geodesics are curves of geodesic curvature 0, both can be considered as particular cases of curves of constant geodesic curvature \lambda, 0 <= |\lambda| <= 1. Thus if convexity is defined with respect to horocycles this limit is 1 and when convexity is defined with respect to geodesics the limit F/L is less or equal than 1. Hence it is natural to ask about the influence of \lambda upon this limit. In fact, when convexity is defined with respect to \lambda-geodesics (equidistants), for each \alpha in [\lambda, 1], there exists a sequence of \lambda convex domains {Kn} expanding over the whole hyperbolic plane such that limn --> \inftyFn/Ln=\alpha. If the sequence is formed by sets with piecewise C2 boundary, then the limsup and liminf of these ratios lie between \lambda and 1.
Another interesting invariant for convex sets in the hyperbolic plane is its diameter D. The question of whether the asymptotic value of D/L is zero or not already appeared in Santaló-Yañez work. We show that for every \mu in [0, 1/2] there is a sequence (Kn) of hyperbolic convex sets such that limD/L = \mu. It must be noticed that in the euclidean case the situation is quite different: any convex set satisfies 1/\pi <= D/L <= 1/2. The lower bound is reached only by constant width sets and the upper bound by segments.
Higher dimensions
The results concerning the ratio perimeter/area of convex sets can be generalized to manifolds of bounded negative curvature, Hadamard manifolds. For sequences of convex sets expanding over the whole hyperbolic space Hn+1 the limit of the quotient volume domain/volume of the boundary is less or equal than 1/n, and exactly 1/n when the sets considered are convex with respect to horocycles. When convexity is with respect to equidistant lines, i.e. curves with constant geodesic curvature \lambda less than one, the above limit has \lambda/n as lower bound. Looking how the boundary bends, we give bounds of the above quotient for a sequence of compact \lambda-convex domains in a complete simply-connected manifold of negative sectional curvature K with -k22 <= K <= -k12: the value of volume domain/volume of the boundary lies between \lambda/n k22 and 1/n k1.
References
Date received: June 20, 2000
Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafh-32.