Two problems on h -convex sets in the hyperbolic space

Abstract

If Q is a convex body in the euclidean space and if \( Q(\varrho) \) denotes the parallel body to distance \( \varrho \) from Q, the limit of \( {V(\varrho) \over F(\varrho)} \) as \( \varrho \) tends to \( \infty \) is known, where \( V(\varrho) \) and \( F(\varrho) \) are respectively the volume of \( Q(\varrho) \) and the area of its boundary.¶When Q is a h-convex body in the hyperbolic plane this problem was analysed by L. A. Santalo, [7] showing that the limits is 1.¶In this paper we study the same question for the hyperbolic space H2n. For our proof the explicit expression of the Isoperimetric Inequality in H2n is essential, as well as the generalized Gauss-Bonnet Theorem, [6].¶We also give the density for the set of horospheres in the n-dimensional hyperbolic space. This density allows us to obtain the measure of the set of horospheres which intersect a compact h-convex hypersurface; equivalently, the volume of the hypersurface can be obtained if we know that measure.