Asymptotics of convex sets in Euclidean and hyperbolic spaces

We study convex sets C of finite (but non-zero) volume in Hn and En. We show that the intersection C∞ of any such set with the ideal boundary of Hn has Minkowski (and thus Hausdorff) dimension of at most (n−1)/2, and this bound is sharp, at least in some dimensions n. We also show a sharp bound when C∞ is a smooth submanifold of ∂∞Hn. In the hyperbolic case, we show that for any k⩽(n−1)/2 there is a bounded section S of C through any prescribed point p, and we show an upper bound on the radius of the ball centered at p containing such a section. We show similar bounds for sections through the origin of a convex body in En, and give asymptotic estimates as 1≪k≪n.