Agustí Reventós

Asymptotic Behaviour of λ-Convex Sets in the Hyperbolic Plane

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 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.

An Interesting Property of the Evolute

where k ? k(s) > 0 is the curvature function of C and ds signifies arclength measure on C. Equality holds if and only if C is a circle. In [1], two proofs of this result are given: the first uses a polygonal approximation of the curve C; the second is based on ideas of Osserman [4]. In this note we give a very short new proof of (1), which has the advantage of providing a geometric interpretation of the difference 2F ? fck~[ds. To be precise, we prove that

Lie flows of codimension 3

We study the following realization problem: given a Lie algebra of dimension 3 and an integer q , 0 < q < 3 , is there a compact manifold endowed with a Lie flow transversely modeled on ¿T and with structural Lie algebra of dimension q ? We give here a quite complete answer to this problem but some questions remain still open (cf. §2). 0. Introduction Among the class of foliations with a transverse structure, Lie foliations stand out. These are foliations transversely modeled on Lie groups. They have been studied by several authors, mainly by Fedida (cf. [3]). Apart from its intrinsic interest, the importance of this study is increased by the fact that they arise naturally in Molinos classification of Riemannian foliations [6]. To each Lie foliation are associated two Lie algebras, the Lie algebra & of the Lie group on which it is modeled and the structural Lie algebra %?. The latter algebra is the Lie algebra of the Lie foliation & restricted to the closure of any one of its leaves. In particular, it is a subalgebra of 9. We remark that although %? is canonically associated to &~, & is not. Thus, one natural and interesting question is to know which pairs of Lie algebras (&, %?), with %? a subalgebra of &, can arise as transverse algebra and structural Lie algebra, respectively, of a Lie foliation y on a compact manifold M. We shall study here a particular but interesting case; namely, given a Lie algebra of dimension 3 and an integer q, 0 < q < 3, is there a compact manifold endowed with a Lie flow transversely modeled on S? and with structural Lie algebra of dimension q ? For simplicitys sake we shall say that the pair (f§, q) is (or is not) realizable. By using the classification of the 3-dimensional Lie algebras and the fact that the structural Lie algebra of a Lie flow is abelian (cf. [1]) it becomes apparent that certain pairs (3?, q) are not realizable (for instance, (sl(2), 2) and (so(3), 2) are not realizable because sl(2) and so(3) have no abelian subalgebras of dimension two). Received by the editors April 14, 1989 and, in revised form, July 6, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 53C12. ©1991 American Mathematical Society 0002-9947/91

A lower bound for the isoperimetric deficit

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A note on Hurwitz's inequality☆

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Evolutes and isoperimetric deficit in two-dimensional spaces of constant curvature

In this paper we provide a Bonnesen-style inequality which gives a lower bound for the isoperimetric deficit corresponding to a closed convex curve in terms of some geometrical invariants of this curve. Moreover we give a geometrical interpretation for the case when equality holds.

On the number of periodic orbits of continuous mappings of the interval

Abstract Given a simple closed plane curve Γ of length L enclosing a compact convex set K of area F , Hurwitz found an upper bound for the isoperimetric deficit, namely L 2 − 4 π F ≤ π | F e | , where F e is the algebraic area enclosed by the evolute of Γ. In this note we improve this inequality finding strictly positive lower bounds for the deficit π | F e | − Δ , where Δ = L 2 − 4 π F . These bounds involve either the visual angle of Γ or the pedal curve associated to K with respect to the Steiner point of K or the L 2 distance between K and the Steiner disk of K . For compact convex sets of constant width Hurwitzs inequality can be improved to L 2 − 4 π F ≤ 4 9 π | F e | . In this case we also get strictly positive lower bounds for the deficit 4 9 π | F e | − Δ . For each established inequality we study when equality holds. This occurs for those compact convex sets being bounded by a curve parallel to an hypocycloid of 3, 4 or 5 cusps or the Minkowski sum of this kind of sets.

Asymptotic Behaviour of ?-Convex Sets in the Hyperbolic Plane

We relate the total curvature and the isoperimetric deficit of a curve

Width of convex bodies in spaces of constant curvature

\gamma

On the structure of the set of periodic points of a continuous map of the interval with finitely many periodic points

in a two-dimensional space of constant curvature with the area enclosed by the evolute of