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

A. A. Borisenko, E. Gallego, A. Reventós

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15 Citations (Scopus)


It is known that for a sequence {Ωt} of convex sets expanding over the whole hyperbolic space Hn+1 the limit of the quotient vol(Ωt)/vol(∂Ωt) 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(Ωt)/vol(∂Ωt) for sequences of λ-convex domains expanding over the whole space lies between the values λ/nk22 and 1/nk1. © 2001 Elsevier Science B.V.
Original languageEnglish
Pages (from-to)267-280
JournalDifferential Geometry and its Application
Publication statusPublished - 1 May 2001


  • 52A10
  • 52A55
  • Hadamard manifold
  • Horocycle
  • Hyperbolic space
  • Normal curvature
  • Volume
  • λ-convex set
  • λ-geodesic


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