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The Cramer-Rao inequality for star bodies. (English) Zbl 1021.52008

Let \(K\) be a convex body in Euclidean space \(R^n\) and let \(o\) be the center of mass of \(K\). Recall that the Legendre ellipsoid \(\Gamma_2 K\) of \(K\) is the ellipsoid centered at \(o\) whose moment of inertia about any axis passing through \(o\) is equal to the corresponding moment of inertia of \(K\).
In Duke Math. J. 104, 375-390 (2000; Zbl 0974.52008)] the authors defined another ellipsoid \(\Gamma_{-2} K\). In the present paper this definition is extended; \(K\) is allowed to be a star body. The main result says that for every convex body \(K\) we have \(\Gamma_{-2} K \subseteq\Gamma_2 K\) with equality if and only if \(K\) is an ellipsoid centered at \(o\). The authors explain that this inequality is the geometric analogue of the Cramer-Rao inequality, which is one of the basic inequalities of information theory.

MSC:

52A40 Inequalities and extremum problems involving convexity in convex geometry
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
94A17 Measures of information, entropy
52A38 Length, area, volume and convex sets (aspects of convex geometry)
52A39 Mixed volumes and related topics in convex geometry

Citations:

Zbl 0974.52008
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References:

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