Stability Inequalities for Projections of Convex Bodies

Abstract

The projection function $P_K$ of an origin-symmetric convex body K in ${{\mathbb {R}}}^n$ is defined by $P_K(\xi )=|K\vert {\xi ^\bot }|,\ \xi \in S^{n-1},$ where $K\vert {\xi ^\bot }$ is the projection of K to the central hyperplane $\xi ^\bot$ perpendicular to $\xi$, and |K| stands for volume of proper dimension. We prove several stability and separation results for the projection function. For example, if D is a projection body in ${{\mathbb {R}}}^n$ which is in isotropic position up to a dilation, and K is any origin-symmetric convex body in ${{\mathbb {R}}}^n$ such that that there exists $\xi \in S^{n-1}$ with $P_K(\xi )>P_D(\xi ),$ then

$$\begin{aligned} \max _{\xi \in S^{n-1}} (P_K(\xi )-P_D(\xi )) \ge \frac{c}{\log ^2n} \big (|K|^{\frac{n-1}{n}} -|D|^{\frac{n-1}{n}}\big ), \end{aligned}$$

where c is an absolute constant. As a consequence, we prove a hyperplane inequality

$$\begin{aligned} S(D) \le \ C \log ^2n \max _{\xi \in S^{n-1}} S(D\vert \xi ^\bot )\ |D|^{\frac{1}{n}}, \end{aligned}$$

where D is a projection body in isotropic position, up to a dilation, S(D) is the surface area of $D,S(D\vert \xi ^\bot )$ is the surface area of the body $D\vert \xi ^\bot$ in ${{\mathbb {R}}}^{n-1},$ and C is an absolute constant. The proofs are based on the Fourier analytic approach to projections developed in [12].