The Cramer-Rao inequality for star bodies (Q1847906)

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.