INEQUALITIES FOR CHORD POWER INTEGRALS

Abstract

For convex bodies, chord power integrals were introduced and studied in several papers (see [3], [6], [14], [15], etc.). The aim of this article is to study them further, that is, we establish the BrunnMinkowski-type inequalities and get the upper bound for chord power integrals of convex bodies. Finally, we get the famous Zhang projection inequality as a corollary. Here, it is deserved to mention that we make use of a completely distinct method, that is using the theory of inclusion measure, to establish the inequality. 1. Preliminaries The setting for this paper is n-dimensional Euclidean space R. We will denote by convex figure a compact convex subset of R, and by convex body a convex figure with nonempty interior. Let Sn−1 denote the unit sphere centered at the origin o in R, and write αn−1 for the (n − 1)-dimensional volume of Sn−1. Let Bn be the closed unit ball in R, write ωn for the n-dimensional volume of Bn. Note that ωn = 2π n 2 nΓ(n2 ) , αn−1 = nωn. By a direction, we mean a unit vector, that is, an element of Sn−1. If u is a direction, we denote by u⊥ the (n − 1)-dimensional subspace orthogonal to u and by lu the line through the origin parallel to u. Denote by AGi,n the affine Grassmann manifold of i-dimensional planes in R. It is a homogeneous space under the action of the motion group G(n) (see [7], p.199). Let dξk be the normalized invariant measure of AGi,n whose restriction to the Grassmann manifold Gi,n is the invariant probability measure. Let ξ1 be a random line intersecting K. Then vol1(K ∩ ξ1) is the chord length of the intersection K ∩ ξ1. The chord power integrals of K are defined by Iλ(K) = 2αn−1 n ∫ ξ1∈AG1,n vol1(K ∩ ξ1)dξ1, 0 ≤ λ < ∞. Received November 30, 2006; Revised April 5, 2007. 2000 Mathematics Subject Classification. 52A40.