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Isotropic measures and maximizing ellipsoids: Between John and Loewner


Authors: Shiri Artstein-Avidan and David Katzin
Journal: Proc. Amer. Math. Soc. 146 (2018), 5379-5390
MSC (2010): Primary 52A40, 52A05, 28A75
DOI: https://doi.org/10.1090/proc/14180
Published electronically: August 14, 2018
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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.


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Additional Information

Shiri Artstein-Avidan
Affiliation: School of Mathematical Sciences, Tel Aviv University, Tel Aviv 6997801, Israel
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

DOI: https://doi.org/10.1090/proc/14180
Keywords: Convex bodies, ellipsoids, John position, Loewner position, isotropic position, M position
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
Article copyright: © Copyright 2018 American Mathematical Society

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