|Title||Isotropic measures and maximizing ellipsoids : Between John and Loewner|
|Author(s)||Artstein-Avidan, Shiri; Katzin, David|
|Source||Proceedings of the American Mathematical Society 146 (2018)12. - ISSN 0002-9939 - p. 5379 - 5390.|
|Publication type||Refereed Article in a scientific journal|
|Keyword(s)||Convex bodies - Ellipsoids - Isotropic position - John position - Loewner position - M position|
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.