Isotropic measures and maximizing ellipsoids: Between John and Loewner

Abstract:

We define a one-parametric family of positions of a centrally symmetric convex body $K$ which interpolates between the John position and the Loewner position: for $r>0$, we say that $K$ is in maximal intersection position of radius $r$ if $\textrm {Vol}_{n}(K\cap rB_{2}^{n})\geq \textrm {Vol}_{n}(K\cap rTB_{2}^{n})$ for all $T\in \rm {SL}_{n}$. We show that under mild conditions on $K$, each such position induces a corresponding isotropic measure on the sphere, which is simply the normalized Lebesgue measure on $r^{-1}K\cap S^{n-1}$. In particular, for $r_{M}$ satisfying $r_{M}^{n}\kappa _{n}=\textrm {Vol}_{n}(K)$, the maximal intersection position of radius $r_{M}$ is an $M$-position, so we get an $M$-position with an associated isotropic measure. Lastly, we give an interpretation of John’s theorem on contact points as a limit case of the measures induced from the maximal intersection positions.