Eduardo Gallego

Relation between area and volume for λ-convex sets in Hadamard manifolds☆

Abstract It is known that for a sequence {Ω t } of convex sets expanding over the whole hyperbolic space H n+1 the limit of the quotient vol (Ω t )/ vol (∂Ω t ) 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 λ less than one, the above limit has λ/n as lower bound. Looking how the boundary bends, in this paper we give bounds of the above quotient for a compact λ -convex domain in a complete simply-connected manifold of negative and bounded sectional curvature, a Hadamard manifold. Then we see that the limit of vol (Ω t )/ vol (∂Ω t ) for sequences of λ -convex domains expanding over the whole space lies between the values λ/nk 2 2 and 1/nk 1 .

The Gauss-Bonnet theorem and Crofton-type formulas in complex space forms

We compute the measure with multiplicity of the set of complex planes intersecting a compact domain in any complex space form. The result is given in terms of the so-called hermitian intrinsic volumes. Moreover, we obtain two different versions for the Gauss-Bonnet-Chern formula in complex space forms. One of them gives the Gauss curvature integral in terms of the Euler characteristic, and some hermitian intrinsic volumes. The other one, which is shorter, involves the measure of complex hyperplanes meeting the domain. As a tool, we obtain variation formulas in integral geometry of complex space forms.

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.

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

Integral geometry and valuations

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A note on Hurwitz's inequality☆

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Integral geometry and geometric inequalities in hyperbolic space

Part I: New Structures on Valuations and Applications.- Translation invariant valuations on convex sets.- Valuations on manifolds.- Part II: Algebraic Integral Geometry.- Classical integral geometry.- Curvature measures and the normal cycle.- Integral geometry of euclidean spaces via Alesker theory.- Valuations and integral geometry on isotropic manifolds.- Hermitian integral geometry.

Asymptotic Behaviour of ?-Convex Sets in the Hyperbolic Plane

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.