# Diameter of a convex body relative to its Legendre ellipsoid

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?

• 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. – Deane Yang Oct 11 '15 at 17:12