The isomorphic section-projection problem for convex bodies

Let $K,L$ be convex bodies in ${\mathbb{R}}^n$ with $K$ centered. Assume that $\mathrm{\mid }K\cap {\theta }^{\perp }\mathrm{\mid }\le \mathrm{\mid }L\mathrm{\mid }{\theta }^{\perp }\mathrm{\mid }$ for all $\theta \in S^{n-1}$. We prove that $\mathrm{\mid }K\mathrm{\mid }\le c\sqrt{n}\mathrm{\mid }L\mathrm{\mid }$, 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.