Inequalities for the difference body of a convex body

In the following, S will denote the boundary of the unit ball in En, and u a variable point of S (so u is a unit vector, or direction). The polar equation of the boundary of DK is given by p=p(u), uCS, so p(u) is the radius of DK in the direction u. Then p(u) is the maximum length of a chord of K having direction u-the length of a diameter of K having direction u. Let,u denote n-dimensional Lebesque measure in E,. Then Rogers and Shephard [5] proved that