The isomorphic section-projection problem for convex bodies

Abstract

Let $K,L$ be convex bodies in ${\mathbb{R}}^n$ with $K$ centered. Assume that $|K\cap \hat{I}{¸}^{\perp }|\le |L|\hat{I}{¸}^{\perp }|$ for all $\hat{I}¸\in S^{n-1}$. We prove that $|K|\le c\sqrt{n}|L|$, which is sharp up to the choice of the absolute constant. The result gives the sharp isomorphic order in a mixed section-projection comparison problem, complementing the isomorphic Busemann-Petty and Shephard problems. It also removes the John's position assumption from an earlier result of the author, up to an absolute constant factor.