Horospheres and convex bodies in \(n\)-dimensional hyperbolic space (Q1430023)

In \(\mathbf{E}^n\) the measure of hyperplanes intersecting a convex domain is proportional to the \((n-2)\)-mean curvature integral of its boundary. In non-Euclidean geometry the totally geodesic hypersurfaces are not always the best analogue to linear hyperplanes. Sometimes horospheres play the role of Euclidean hyperplanes. In hyperbolic space of dimension \(n=2\) and \(3\) Santaló proved that the measure of horospheres intersecting a convex domain is proportional to the \((n-2)\)-mean curvature integral of its boundary. The authors show that this does not generalize for higher dimensional hyperbolic spaces, in particular, they express the measure of horospheres intersecting a convex body as a linear combination of the mean curvature integrals of its boundary.