Abstract
A question related to some conjectures of Lutwak about the affine quermassintegrals of a convex body K in Rn asks whether for every convex body K in Rn and all 1⩽k⩽nΦ[k](K):=voln(K)−1n(∫Gn,kvolk(PF(K))−ndνn,k(F))−1kn⩽cn/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 Φ[k](K) when K=B1n, the unit ball of ℓ1n, and explain how this special instance has implications for the case of a general unconditional convex body K.
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