Isomorphic Busemann--Petty for arbitrary measures: the sharp order

Abstract

Let $C_n$ be the optimal constant with the following property. For every even, continuous, strictly positive density $f$ on $R^n$ and all origin-symmetric convex bodies $K,L\subset R^n$, the inequalities $${\int }_{K\cap \hat{I}{¾}^{\perp }}f\le {\int }_{L\cap \hat{I}{¾}^{\perp }}f\text{for all }\hat{I}¾\in S^{n-1}$$ imply ${\int }_{K}f\le C_n{\int }_{L}f$. In an earlier paper the authors proved that $C_n\le \sqrt{n}$. In this paper, we prove the matching lower bound $C_n\ge c\sqrt{n}$. To simplify the exposition, we first give a complete one-scale construction, based on earlier work of Klartag and Koldobsky, which yields $C_n\ge c\sqrt{n/\log n}$. For the sharp result, we use the random-rounding construction of Klartag and Livshyts as a black box and combine it with a spherical-averaging support-separation argument.