Isotropic Measures and Maximizing Ellipsoids: Between John and Loewner

Abstract

We define a one parameter family of positions of a convex body 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 ${\text{Vol}}_n(K\cap r{B}_2^n)\ge {\text{Vol}}_n(K\cap rT{B}_2^n)$ for all $T\in S{L}_n$. We show that under mild conditions on $K$, each such position induces a corresponding isotropic measure on the sphere, which is simply a normalized Lebesgue measure on $r^{-1}K\cap S^{n-1}$. In particular, for $r_M$ satisfying $r_M^n{\kappa }_n={\text{Vol}}_n(K)$, 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.

10 pages

References in corpus (2)

• On some inequalities for Gaussian measures
• More on logarithmic sums of convex bodies

Cited by in corpus (1)

• On explicit representations of isotropic measures in John and Löwner positions