Associated with each body $K$ in Euclidean $n$-space $\mathbb
{R}\sp n$ is an ellipsoid $\Gamma\sb 2K$ 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 $K\subset\mathbb {R}\sp n$ is a new ellipsoid $\Gamma\sb
{-2}K$ 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 $\Gamma\sb {-2}K$ is the corresponding object
of the Brunn-Minkowski theory.
¶ The present paper has two aims. The first is to show that the
domain of $\Gamma\sb {-2}$ 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
$\Gamma\sb {-2}K\subset\Gamma\sb 2K$
¶ 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.