Horospheres and convex bodies in n-dimensional hyperbolic space

E. Gallego, A. M. Naveira, G. Solanes

Research output: Contribution to journalArticleResearchpeer-review

4 Citations (Scopus)

Abstract

In n-dimensional Euclidean space, the measure of hyperplanes intersecting a convex domain is proportional to the (n - 2)-mean curvature integral of its boundary. This question was considered by Santaló in hyperbolic space. In non-Euclidean geometry the totally geodesic hypersurfaces are not always the best analogue to linear hyperplanes. In some situations horospheres play the role of Euclidean hyperplanes. In dimensions n = 2 and 3, Santaló proved that the measure of horospheres intersecting a convex domain is also proportional to the (n - 2)-mean curvature integral of its boundary. In this paper we show that this analogy does not generalize to higher dimensions. We express the measure of horospheres intersecting a convex body as a linear combination of the mean curvature integrals of its boundary.
Original languageEnglish
Pages (from-to)103-114
JournalGeometriae Dedicata
Volume103
Issue number1
DOIs
Publication statusPublished - 1 Feb 2004

Keywords

  • Convex set
  • H-convex set
  • Horocycle
  • Horosphere
  • Hyperbolic space
  • Volume

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