Intrinsic volumes and polar sets in spherical space

Abstract. For a convex body of given volume in spherical space, the total invariant measure of hitting great subspheres becomes minimal, equivalently the volume of the polar body becomes maximal, if and only if the body is a spherical cap. This result can be considered as a spherical counterpart of two Euclidean inequalities, the Urysohn inequality connecting mean width and volume, and the Blaschke-Santaló inequality for the volumes of polar convex bodies. Two proofs are given; the first one can be adapted to hyperbolic space. MSC 2000: 52A40 (primary); 52A55, 52A22 (secondary) Keywords: spherical convexity, Urysohn inequality, BlaschkeSantaló inequality, intrinsic volume, quermassintegral, polar convex bodies, two-point symmetrization In the classical geometry of convex bodies in Euclidean space, the intrinsic volumes (or quermassintegrals) play a central role. These functionals have counterparts in spherical and hyperbolic geometry, and it is a challenge for a geometer to find analogues in these spaces for the known results about Euclidean intrinsic volumes. In the present paper, we prove one such result, an inequality of isoperimetric type in spherical (or hyperbolic) space, which can be considered as a counterpart to the Urysohn inequality and is, in the spherical case, also related to the Blaschke-Santaló inequality. Before stating the result (at the beginning of Section 2), we want to recall and collect the basic facts about intrinsic volumes in Euclidean space, describe their noneuclidean counterparts, and state those open problems about the latter which we consider to be of particular interest. We restrict ourselves here to spherical space and comment only briefly on the case of hyperbolic space