The Cramer-rao Inequality for Star Bodies

Associated with each body K in Euclidean n-space Rn is an ellipsoid 02K called the Legendre ellipsoid of K . It can be defined as the unique ellipsoid centered at the body’s center of mass such that the ellipsoid’s moment of inertia about any axis passing through the center of mass is the same as that of the body. In an earlier paper the authors showed that corresponding to each convex body K ⊂ Rn is a new ellipsoid 0−2K that is in some sense dual to the Legendre ellipsoid. The Legendre ellipsoid is an object of the dual Brunn-Minkowski theory, while the new ellipsoid 0−2K is the corresponding object of the Brunn-Minkowski theory. The present paper has two aims. The first is to show that the domain of 0−2 can be extended to star-shaped sets. The second is to prove that the following relationship exists between the two ellipsoids: If K is a star-shaped set, then 0−2K ⊂ 02K with equality if and only if K is an ellipsoid centered at the origin. This inclusion is the geometric analogue of one of the basic inequalities of information theory—the Cramer-Rao inequality. Associated with each body K in Euclidean n-space Rn is an ellipsoid 02K called the Legendre ellipsoid of K . The Legendre ellipsoid is a basic concept from classical mechanics. It can be defined as the unique ellipsoid centered at the body’s center of mass such that the ellipsoid’s moment of inertia about any axis passing through the center of mass is the same as that of the body. In [26] the authors showed that corresponding to each convex body K ⊂ Rn is a new ellipsoid 0−2K . The results in this paper hint at a remarkable duality between this new ellipsoid and the Legendre ellipsoid. DUKE MATHEMATICAL JOURNAL Vol. 112, No. 1, c © 2002 Received 12 September 2000. Revision received 31 March 2001. 2000 Mathematics Subject Classification. Primary 52A40; Secondary 94A17. Author’s work supported in part by National Science Foundation grant numbers DMS-9803261 and DMS0104363