Affine quermassintegrals of random polytopes

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 $1leqslant kleqslant n$ $$Phi_{[k]}(K):={rm vol}_n(K)^{-frac{1}{n}}left (int_{G_{n,k}}{rm vol}_k(P_F(K))^{-n},d u_{n,k}(F)right )^{-frac{1}{kn}}leqslant csqrt{n/k},$$ 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 $K$.