Isotropic measures and maximizing ellipsoids: Between John and Loewner

Shiri Artstein-Avidan, David Katzin

Research output: Contribution to journalArticleAcademicpeer-review

1 Citation (Scopus)

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 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.

Original languageEnglish
Pages (from-to)5379-5390
JournalProceedings of the American Mathematical Society
Volume146
Issue number12
DOIs
Publication statusPublished - 1 Dec 2018

Keywords

  • Convex bodies
  • Ellipsoids
  • Isotropic position
  • John position
  • Loewner position
  • M position

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