Isotropic measures and maximizing ellipsoids: Between John and Loewner
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- by Shiri Artstein-Avidan and David Katzin PDF
- Proc. Amer. Math. Soc. 146 (2018), 5379-5390 Request permission
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.References
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Additional Information
- Shiri Artstein-Avidan
- Affiliation: School of Mathematical Sciences, Tel Aviv University, Tel Aviv 6997801, Israel
- MR Author ID: 708154
- Email: shiri@post.tau.ac.il
- David Katzin
- Affiliation: School of Mathematical Sciences, Tel Aviv University, Tel Aviv 6997801, Israel
- Address at time of publication: Wageningen University, Droevendaalsesteeg 1, 6708 PB Wageningen, The Netherlands
- Email: david.katzin@wur.nl
- Received by editor(s): July 24, 2017
- Received by editor(s) in revised form: April 2, 2018
- Published electronically: August 14, 2018
- Additional Notes: The authors were supported in part by ISF grant No. 665/15.
- Communicated by: Thomas Schlumprecht
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 5379-5390
- MSC (2010): Primary 52A40, 52A05, 28A75
- DOI: https://doi.org/10.1090/proc/14180
- MathSciNet review: 3866876