Horospheres and Convex Bodies in n-Dimensional Hyperbolic Space |
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Authors: | E. Gallego A. M. Naveira G. Solanes |
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Affiliation: | (1) Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain;(2) Facultad de Matemáticas, Departamento de Geometría y Topología, Avenida Andrés Estellés 1, 46100 Burjassot (València), Spain |
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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. |
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Keywords: | convex set h-convex set horocycle horosphere hyperbolic space volume |
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