Resumen de Asymptotics of convex sets in Euclidean and hyperbolic spaces

Igor Rivin

• We study convex sets C of finite (but non-zero) volume in and . We show that the intersection C8 of any such set with the ideal boundary of 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 C8 is a smooth submanifold of . 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 , and give asymptotic estimates as 1kn.