The projection function PK\documentclass[12pt]{minimal}
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\begin{document}$$P_K$$\end{document} of an origin-symmetric convex body K in Rn\documentclass[12pt]{minimal}
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\begin{document}$${{\mathbb {R}}}^n$$\end{document} is defined by PK(ξ)=|K|ξ⊥|,ξ∈Sn-1,\documentclass[12pt]{minimal}
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\begin{document}$$P_K(\xi )=|K\vert {\xi ^\bot }|,\ \xi \in S^{n-1},$$\end{document} where K|ξ⊥\documentclass[12pt]{minimal}
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\begin{document}$$K\vert {\xi ^\bot }$$\end{document} is the projection of K to the central hyperplane ξ⊥\documentclass[12pt]{minimal}
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\begin{document}$$\xi ^\bot $$\end{document} perpendicular to ξ\documentclass[12pt]{minimal}
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\begin{document}$$\xi $$\end{document}, 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 Rn\documentclass[12pt]{minimal}
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\begin{document}$${{\mathbb {R}}}^n$$\end{document} which is in isotropic position up to a dilation, and K is any origin-symmetric convex body in Rn\documentclass[12pt]{minimal}
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\begin{document}$${{\mathbb {R}}}^n$$\end{document} such that that there exists ξ∈Sn-1\documentclass[12pt]{minimal}
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\begin{document}$$\xi \in S^{n-1}$$\end{document} with PK(ξ)>PD(ξ),\documentclass[12pt]{minimal}
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\begin{document}$$P_K(\xi )>P_D(\xi ),$$\end{document} then maxξ∈Sn-1(PK(ξ)-PD(ξ))≥clog2n(|K|n-1n-|D|n-1n),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \max _{\xi \in S^{n-1}} (P_K(\xi )-P_D(\xi )) \ge \frac{c}{\log ^2n} \big (|K|^{\frac{n-1}{n}} -|D|^{\frac{n-1}{n}}\big ), \end{aligned}$$\end{document}where c is an absolute constant. As a consequence, we prove a hyperplane inequality S(D)≤Clog2nmaxξ∈Sn-1S(D|ξ⊥)|D|1n,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} S(D) \le \ C \log ^2n \max _{\xi \in S^{n-1}} S(D\vert \xi ^\bot )\ |D|^{\frac{1}{n}}, \end{aligned}$$\end{document}where D is a projection body in isotropic position, up to a dilation, S(D) is the surface area of D,S(D|ξ⊥)\documentclass[12pt]{minimal}
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\begin{document}$$D,S(D\vert \xi ^\bot )$$\end{document} is the surface area of the body D|ξ⊥\documentclass[12pt]{minimal}
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\begin{document}$$D\vert \xi ^\bot $$\end{document} in Rn-1,\documentclass[12pt]{minimal}
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\begin{document}$${{\mathbb {R}}}^{n-1},$$\end{document} and C is an absolute constant. The proofs are based on the Fourier analytic approach to projections developed in [12].
