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arXiv · 2111.03624

On explicit representations of isotropic measures in John and L\"owner positions

F. M. Baêta·J. Haddad
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Abstract

Given a convex body $K \subseteq \mathbb R^n$ in L\"owner position we study the problem of constructing a non-negative centered isotropic measure supported in the contact points, whose existence is guaranteed by John's Theorem. The method we propose requires the minimization of a convex function defined in an $\frac {n(n+3)}2$ dimensional vector space. We find a geometric interpretation of the minimizer as $\left. \frac{\partial}{\partial r}(A_r, v_r)\right|_{r=1}$, where $A_r K + v_r$ is a one-parameter family of positions of $K$ that are in some sense related to the maximal intersection position of radius $r$ defined recently by Artstein-Avidan and Katzin.

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F. M. Baêta, J. Haddad. 2021-11-05. On explicit representations of isotropic measures in John and L\"owner positions. https://doi.org/10.1093/imrn%2Frnac269

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