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.
|Publication status||Published - 1 Feb 2004|
- Convex set
- H-convex set
- Hyperbolic space