Measure comparison problems for dilations of convex bodies
Malak Lafi, Artem Zvavitch
TL;DR
The paper investigates dilation-based analogues of the Busemann-Petty problem for log-concave measures on $\mathbb{R}^n$, asking whether comparing measures of dilates or sections across all directions and scales implies a containment or a measure order. It develops a norm-adapted large-deviation framework and proves that for rotation-invariant densities with density $e^{-\phi(|x|)}$, the implication $\mu(tRB_2^n)\le \mu(tK)$ for large $t$ forces $RB_2^n\subset K$, with a generalization to densities $e^{-\phi(\|x\|_{L})}$ via $r(K,L)=\max\{R: RL\subset K\}$. Despite these positive results, the paper also establishes negative answers to the dilation version of the Busemann-Petty problem in high dimensions: for $n\ge 5$ (and $n\ge 7$ under rotation invariance) there exist $K,L$ and a log-concave $\mu$ such that $\mu(rK\cap \xi^{\perp})\le \mu(rL\cap \xi^{\perp})$ for all $\xi,r$, yet $\mu(K)>.\mu(L)$. These findings illuminate sharp dimension thresholds and connect Gaussian, general log-concave, and norm-dependent measures, while introducing a versatile large-deviation approach for convex-geometry problems under measures.
Abstract
We study a version of the Busemann-Petty problem for $\log$-concave measures with an additional assumption on the dilates of convex, symmetric bodies. One of our main tools is an analog of the classical large deviation principle applied to $\log$-concave measures, depending on the norm of a convex body. We hope this will be of independent interest.