Affine quermassintegrals of random polytopes

Abstract: A question related to some conjectures of Lutwak about the affine quermassintegrals of a convex body $K$ in ${\mathbb R}^n$ asks whether for every convex body $K$ in ${\mathbb R}^n$ and all $1\leqslant k\leqslant n$ $$\Phi_{[k]}(K):={\rm vol}n(K)^{-\frac{1}{n}}\left (\int{G_{n,k}}{\rm vol}k(P_F(K))^{-n}\,d\nu{n,k}(F)\right )^{-\frac{1}{kn}}\leqslant c\sqrt{n/k},$$ where $c>0$ is an absolute constant. We provide an affirmative answer for some broad classes of random polytopes. We also discuss upper bounds for $\Phi_{[k]}(K)$ when $K=B_1^n$, the unit ball of $\ell_1^n$, and explain how this special instance has implications for the case of a general unconditional convex body $K$.