Isotropic measures and maximizing ellipsoids: Between John and Loewner

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 Vol _n (K ∩ rB ⁿ ₂ ) ≥ Vol _n (K ∩ rTB ⁿ ₂ ) for all T ∈ 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 ∩ S ⁿ⁻¹ . Inparticular, for r _M satisfying r ⁿ _M κ _n =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.