An isomorphic version of the Busemann-Petty problem for arbitrary measures

Abstract

We prove the following theorem. Let $\mu$ be a measure on $R^n$ with even continuous density, and let $K,L$ be origin-symmetric convex bodies in $R^n$ so that $\mu (K\cap H)\le \mu (L\cap H)$ for any central hyperplane H. Then $\mu (K)\le \sqrt{n}\mu (L).$ We also prove this result with better constants for some special classes of measures and bodies. Finally, we prove a version of the hyperplane inequality for convex measures.

References in corpus (2)

• Complex Intersection Bodies
• A hyperplane inequality for measures of unconditional convex bodies