The Cramer-Rao inequality for star bodies

Research output: Contribution to journalArticle

Abstract

Associated with each body K in Euclidean n-space ℝn is an ellipsoid Γ2K 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 Γ-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 Γ-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 Γ-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 Γ-2K ⊂2K 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.

Original languageEnglish (US)
Pages (from-to)59-81
Number of pages23
JournalDuke Mathematical Journal
Volume112
Issue number1
DOIs
StatePublished - Mar 15 2002

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Cramér-Rao Inequality
Star Body
Ellipsoid
Legendre
Star-shaped Set
Barycentre
Moment of inertia
Convex Body
Information Theory
Euclidean
Equality
Inclusion

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

The Cramer-Rao inequality for star bodies. / Lutwak, Erwin; Yang, Deane; Zhang, Gaoyong.

In: Duke Mathematical Journal, Vol. 112, No. 1, 15.03.2002, p. 59-81.

Research output: Contribution to journalArticle

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