Given a convex body in $\mathbb{R}^n$ that is symmetric with respect to the origin, let us measure its diameter with respect to the Euclidean metric determined by its own Legendre ellipsoid. How large can this diameter be? Is there a known sharp upper bound?

1$\begingroup$ You're essentially trying to bound an "affine diameter" from above by the volume of the body. This is more or less a reverse affine isoperimetric inequality. There are not many known sharp inequalities of this type. $\endgroup$ – Deane Yang Oct 11 '15 at 17:12
There are different notions of legendre ellipsoid depending on whether you have a background euclidean structure on the space (and in this case you also have an additional possibily because you may divide say by a volume, or not), or if you have no background euclidean structure and you first construct the legendre ellipsoid on the cotangent space and then take its dual in the tangent space (how it is done in [V. D. Milman, A. Pajor, Lecture Notes in Math., 1376, Springer,1989] or in http://xxx.lanl.gov/abs/1104.1647).
In the second case, there are some estimations in http://xxx.lanl.gov/abs/1408.6401 both in the symmetric and nonsymmetriccase
In the first case, one can get better results using the classically known formulas how compare the norm given by the convex body and by its John and Loevner ellipsoids (some references are also in http://xxx.lanl.gov/abs/1408.6401). I do not remember by hart whether in the symmetric case they have exact estimations.

$\begingroup$ Indeed, the more natural interpretation of the question is the second one, where the Legendre ellipsoid lies in the vector space dual to the one containing the body. $\endgroup$ – Deane Yang Oct 11 '15 at 17:13

$\begingroup$ Aren't you confusing the Binet and the Legendre ellipsoids? The Binet Ellipsoid is in the dual space and the Legendre ellipsoid is its dual, in the same space as the convex body. $\endgroup$ – alvarezpaiva Oct 11 '15 at 18:28


$\begingroup$ Juan Carlos, what is your definition of the legendre ellipsoid? $\endgroup$ – Vladimir S Matveev Oct 12 '15 at 17:02