Perimeter, Diameter and Area of Convex Sets in the Hyperbolic Plane

PERIMETER, DIAMETER AND AREA OF CONVEX SETS IN THE HYPERBOLIC PLANE. EDUARDO GALLEGO AND GIL SOLANES Abstract. In this paper we study 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 (cf. [3] and [2]) that F/L goes to a value between 0 and 1 depending on the shape of the contour. Here, rst of all it is 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/ (D/L = 1/ if and only if the convex set has constant width). Moreover, it is shown that, as the limit of D/L approaches to 1/2, the possible limit values of F/L reduce. Examples of all possible limits F/L and D/L are given. 1. Introduction In hyperbolic geometry, given a point p exterior to a line l there are innitely many non secant lines. These lines lie between the two so called parallel lines to l. When the distance from p to l grows to innity, the angle between the parallel lines goes to 0. This fact leads to the ambiguous idea that, in some sense, given a line l, the probability that a random line meets l is zero. In order to formalize this idea let us restrict our attention to the interiors of a sequence (Kn) of convex sets in the hyperbolic plane expanding to ll it. The probability for a random chord of Kn to meet r inside Kn should go to 0 as n . It can be proved by using the → ∞ Cauchy-Crofton formula, that this probability is 2n/Ln, where n is the length of the chord l Kn and Ln denotes the length of ∂Kn. Because the length of the ∩ chord n is less or equal than the diameter Dn of Kn, the study of lim n/Ln is related to the knowledge of the asymptotic value of the ratio Dn/Ln where Dn. The question of whether the asymptotic value of D/L is zero or not already appeared in [7]. In the present text we will see that there are many possible values for this limit and we will nd them all. More precisely we will prove that for every e [0, 1/2] there is a sequence (Kn) of hyperbolic convex sets such ∈ that lim Dn/Ln = e. In fact it will be seen that, for convex sets with respect to equidistants intersecting innity with angle , this limit can take values only below (sin )/2. It must be noticed that in the euclidean case the situation is quite dierent: any convex set satises 1/ D/L 1/2. The lower bound is reached only by constant width sets and the upper bound by the segments. The paper is organized as follows. In sections 2 and 3 we introduce the basic concepts and notation. In section 4 we nd lower and upper bounds for D/L in the -convex case, concluding that the asymptotic value of D/L for h-convex sequences is 0. Section 5 is devoted to the construction of examples showing that the preceding bounds are the best possible. In section 6 we recall the asymptotic behavior of the quotient F/L being F the area of the convex sets. We introduce, in section 7, the 1991 Mathematics Subject Classication. Primary 52A55; Secondary 52A10. Key words and phrases. Hyperbolic plane, diameter, perimeter, area, -geodesic, -convex set. Work partially supported by DGYCIT grant number PB96 – 1178. 1 2 EDUARDO GALLEGO AND GIL SOLANES metric space of hyperbolic convex sets in order to treat an isoperimetric problem. Finally, in section 8, we give the relation between the asymptotic values of D/L and F/L. More precisely, we can state that F D 2 lim n 1 2 lim n . n L n L →∞ n s →∞ n We wish to thank professor A. Reventosfor many helpful conversations during the preparation of this work. 2. The hyperbolic plane In this section we introduce the hyperbolic plane as well as some basic facts that will be used later on. The hyperbolic plane, H2, is the unique complete simply connected Riemannian manifold of dimension 2 with constant curvature 1. Its geometry corresponds to the one obtained from the absolute geometry given by the rst four Euclid postulates and the Lobachevsky postulate: through every point P exterior to a line l pass more than one line not intersecting l. It is useful to have dierent models for this geometry, we shall describe their points, lines (geodesics) and rigid motions: 2 1 2 Half-plane model. It is the half-plane (x, y) R y > 0 with the metric 2 (dx + { ∈ | } y dy2). The geodesics are half-circles centered in y = 0 and vertical half-lines. The rigid motions are composition of inversions with{ respect} to these circles and symmetries with respect to these lines. This model is conformal since the metric is a multiple of the euclidean metric. 4 2 2 Disk model. It is the unit disk with the metric (1 x2 y2)2 (dx + dy ). This model is also conformal. The geodesics are the a diameters of the disk and the arcs of circumference orthogonal to the border. The rigid motions are homographies of the complex plane xing the disk. 1 1 2 2 2 Projective model. It is the unit disk with the metric 1 r2 ( 1 r2 dr + r d ) where (r, ) are the euclidean polar coordinates centered at the origin. The geodesics are chords of the disk. This fact makes this model become very useful when studying questions related to convex sets. The rigid motions are the projectivities xing the disk. In the following sections polar coordinates will be useful in the treatment of some problems. Whatever it is the model we work in, we can parametrize the points of the hyperbolic plane in the following way. Let O be a point called origin. We 2 choose in O a direction v TOH . For each point P , let r be the length of the geodesic segment joining O∈and P , and let be the angle between this segment and v. Now, H2 O is perfectly parametrized by the coordinates (r, ). It can be easily checked out\{ that} in these coordinates the metric is written as follows. g = dr2 + (sinh r)2d2. The volume element will be then sinh rdrd and the area and perimeter of a cir- 2 cumference of radius r in H are 2 r 2 L = sinh rd = 2 sinh r, F = sinh rddr = 2(cosh r 1). Z0 Z0 Z0 We shall need some formulas in hyperbolic trigonometry; proofs can be found in [5]. Let a, b and c be three sides of a geodesic triangle and let , and be their opposite angles. The following identities are then veried: cosh a = cosh b cosh c sinh b sinh c cos , CONVEX SETS IN THE HYPERBOLIC PLANE 3 sinh a sinh b sinh c = = , sin sin sin sinh a cos = cosh b sinh c sinh b cosh c cos . The area of such triangle is F = ( + + ). It is said that the area equals the angular defect. To end this section we will just state some integral geometric formulas used later on. The isoperimetric inequality (1) L2 4F F 2 0 gives a relation between perimeter L and area F of an arbitrary compact domain in H2. The Cauchy-Crofton theorem expresses the length of a curve in terms of the measure of lines (counted with its multiplicity) intersecting it. More precisely 2L = n(r)dr r C= Z ∩ 6 ∅ where dr is a normalized isometry invariant density of geodesic lines and n(r) the number of intersecting points of r and the curve C (cf. [6]). 3. Convexity and -convexity In this section we introduce the concept of -convexity as well as some basic known facts about it. For a more detailed introduction see [3]. 2 Denition 3.1. A subset K H is said to be convex if for every pair of points in K, the geodesic segment joining them is also in K. Notice that a set in H2 is convex if and only if in the projective model it looks like an euclidean convex set. Denition 3.2. A closed convex set with nonempty interior is called a convex domain. From now on all convex sets will be compact convex domains. If K is a convex set, then ∂K is C2 except from, at most, a countable set of points. Moreover, ∂K must have nite length, which is called the perimeter L of K, and the area F of K must be nite too. The diameter is given by D = max d(p, q) p, q ∂K . { | ∈ } Denition 3.3. A sequence (Kn) of convex sets is said to expand over the whole 2 hyperbolic plane if Kn Kn+1 and p H there is an n such that p Kn. ∀ ∈ ∈ As in the euclidean case we have Lemma 3.1. A compact domain with piecewise C2 boundary is convex if and only if its geodesic curvature does not change the sign and in the non C2 points the interior angles are not greater than . Given a geodesic line l in the euclidean plane, the set of equidistant points to l are two parallel lines symmetric with respect to l. In the hyperbolic plane this is no longer true; the set of equidistant points to l are two smooth curves called equidistants.

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