The John inclusion for log-concave functions
G. Ivanov
TL;DR
The paper extends John’s inclusion from convex geometry to the realm of log-concave functions by formulating a Functional John problem and exploiting polar duality. It proves a John-type inclusion: when hbar is the John function of a proper log-concave f, we have hbar ≤ f and e^{-(d+1)} hbar∘[(d+1)Id_d] ≤ f^∘, culminating in an asymptotically tight inequality χ_{B^d} ≤ α f(A(x-a)) ≤ sqrt{d+1} e^{-|x|/(d+2)+(d+1)}. The work also develops a robust structural theory for John decompositions of identity in the functional setting, reduces to positive John bumps, and establishes sharp height bounds, while showing the nonexistence of a Löwner-type inclusion in this context. These results unify John-type and duality ideas for functions and open questions about the extent and limits of functional dualities and tail behavior. Overall, the paper advances the functional convexity program by providing precise, quantitative, and asymptotically optimal inclusions for log-concave functions and highlighting the role of polar structures in the function space.
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 $α$ 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 $χ_{\mathbf{B}^{d}}$ is the indicator function of the unit ball $\mathbf{B}^{d}$.