The John inclusion for log-concave functions

TL;DR

This paper extends John's inclusion to log-concave functions, establishing a tight inequality relating these functions to the indicator of the unit ball, with implications for convex geometry and analysis.

Contribution

It introduces a John-type inclusion for log-concave functions and proves a tight inequality connecting such functions to the unit ball indicator.

Findings

Established a John-type inclusion for log-concave functions.

Proved an asymptotically tight inequality involving log-concave functions and the unit ball.

Derived bounds involving positive definite matrices and affine transformations.

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 $\alpha$ such that \[ \chi_{\mathbf{B}^{d}}(x) \leq \alpha f\!\!\left(A(x-a)\right) \leq \sqrt{d+1} \cdot e^{-\frac{\left|x\right|}{d+2} + (d+1)}, \] where $\chi_{\mathbf{B}^{d}}$ is the indicator function of the unit ball $\mathbf{B}^{d}$.