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 K which interpolates between the John position and the Loewner position: for r > 0 r>0 , we say that K K is in maximal intersection position of radius r r if Vol n ( K ∩ r B 2 n ) ≥ Vol n ( K ∩ r T B 2 n ) \textrm {Vol}_{n}(K\cap rB_{2}^{n})\geq \textrm {Vol}_{n}(K\cap rTB_{2}^{n}) for all T ∈ S L n T\in \rm {SL}_{n} . We show that under mild conditions on K K , each such position induces a corresponding isotropic measure on the sphere, which is simply the normalized Lebesgue measure on r − 1 K ∩ S n − 1 r^{-1}K\cap S^{n-1} . In particular, for r M r_{M} satisfying r M n κ n = Vol n ( K ) r_{M}^{n}\kappa _{n}=\textrm {Vol}_{n}(K) , the maximal intersection position of radius r M r_{M} is an M M -position, so we get an M 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.