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 H ^{2 n}. For our proof the explicit expression of the Isoperimetric Inequality in H ^{2 n} 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.