On the Comparison of Measures of Convex Bodies via Projections and Sections

Abstract

Abstract We study the inequalities between measures of convex bodies implied by comparison of their projections and sections. Recently, Giannopoulos and Koldobsky proved that if $K, L$ are convex bodies such that the volume of the projection of $K$ onto any hyperplane through the origin is bounded by the volume of the intersection of $L$ with the hyperplane, then $|K| \le |L|$. Firstly, we study the reverse question: in particular, we show that if $K, L$ are origin-symmetric convex bodies in John’s position such that the volume of the intersection of $K$ with any hyperplane through the origin is bounded by the volume of the projection of $L$ onto the hyperplane, then $|K| \le \sqrt {n}|L|$. The condition we consider is weaker than the conditions that appear in the Busemann–Petty and Shephard problems. Secondly, we appropriately extend the result of Giannopoulos and Koldobsky to various classes of measures possessing concavity properties, including log-concave measures.