Regular ellipsoids and a Blaschke-Santal\'o-type inequality for projections of non-symmetric convex bodies

Abstract

It is shown that every not-necessarily symmetric convex body $K$ in ${\mathbb R}^n$ has an affine image $\tilde{K}$ of $K$ such that the covering numbers of $\tilde{K}$ by growing dilates of the unit Euclidean ball, as well as those of the unit Euclidean ball by growing dilates of $\tilde{K}$, decrease in a regular way. This extends to the non-symmetric case a famous theorem by Pisier, albeit with worse estimates on the rate of decrease of the covering numbers. The affine image $\tilde{K}$ can be chosen to have either barycentre or Santal\'o point at the origin. In the proof we use Pisier's theorem as a black box, as well as a suggested approach by Klartag and V. Milman. A key new ingredient is Blaschke-Santal\'{o}-type inequalities for projections of a body $\tilde{K}$ with Santal\'{o} point at the origin, which could be of independent interest. Unlike the application to covering, these (as well as the analogous inequalities for centred convex bodies that were already considered by Klartag and Milman ["Rapid Steiner symmetrization of most of a convex body and the slicing problem'', Comb., Prob. & Comp. 14 (2005), preprint version]) can be shown to be optimal up to absolute constants. We also present an application to results around the mean norm of isotropic (not-necessarily symmetric) convex bodies.