On quermassintegrals of mixed projection bodies (Q583613)

For convex bodies K in \({\mathbb{R}}^ n\), the author discusses the inequalities \[ (P_ i)\quad W_{i+1}(K)^{n-i}\quad \geq \quad [\omega_ n^{n-1-i}/\omega^{n-i}_{n-1}]W_ i(\Pi_ iK)\quad \geq \quad \omega_ nW_ i(K)^{n-i-1} \] (i\(=0,...,n-2)\). Here \(W_ i\) is the ith quermassintegral, \(\omega_ n\) is the volume of the unit ball in \({\mathbb{R}}^ n\), and \(\Pi_ iK\) is the ith projection body of K; this is the body defined by \(h(\Pi_ i,K,u)=w_ i(K^ u)\) for unit vectors u, where h denotes the support function, \(K^ u\) is the image of K under orthogonal projection onto a hyperplane orthogonal to u, and \(w_ i\) is the ith quermassintegral of dimension n-1. The author proves the left inequality in \((P_ i)\) and conjectures the right one, which he proves for \(i=n-2\). \((P_ i)\) would improve the known inequality \(W_{i+1}(K)^{n-1}\geq \omega_ nW_ i(K)^{n-i-1}\). The right side of \((P_ 0)\) is a conjecture of Petty of 1972. It is shown here that the truth of \((P_ 0)\) implies the truth of \((P_ i)\) for \(i=1,...,n-2\).