## 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 language | English |
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Pages (from-to) | 5379-5390 |

Journal | Proceedings of the American Mathematical Society |

Volume | 146 |

Issue number | 12 |

DOIs | |

Publication status | Published - 1 Dec 2018 |

## Keywords

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