The John inclusion for log-concave functions

Abstract

John's inclusion states that a convex body in ${\mathbb{R}}^d$ can be covered by the $d$-dilation of its maximal volume ellipsoid. We obtain a certain John-type inclusion for log-concave functions. As a byproduct of our approach, we establish the following asymptotically tight inequality: \\ \noindent For any log-concave function $f$ with finite, positive integral, there exist a positive definite matrix $A$, a point $a\in {\mathbb{R}}^d$, and a positive constant $\hat{I}\pm$ such that \[ χ_{\mathbf{B}^{d}}(x) \leq αf\!\!\left(A(x-a)\right) \leq \sqrt{d+1} \cdot e^{-\frac{\left|x\right|}{d+2} + (d+1)}, \] where $\ddot{I}{‡}_{{\mathbf{B}}^d}$ is the indicator function of the unit ball ${\mathbf{B}}^d$.