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A problem on the affine quermassintegrals of convex bodies Chasapis, Giorgos

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We study a variant of one of Lutwak's conjectures on the affine quermassintegrals of a convex body: Is it true that \[ \frac{1}{\mathrm{vol}_n(K)^{\frac{1}{n}}} \left(\int_{G_{n,k}} \mathrm{vol}_k(P_F(K))^{-n}\,d\nu_{n,k}(F) \right)^{-\frac{1}{kn}} \leqslant c\sqrt{\frac{n}{k}} \] holds for every convex body $K$ in $\mathbb{R}^n$ and all $1\leqslant k\leqslant n$, for some absolute constant $c>0$ Here integration is with respect to the rotation-invariant probability measure $\nu_{n,k}$ on the Grassmanian $G_{n,k}$ of all $k$-dimensional subspaces of $\mathbb{R}^n$, and $P_F$ denotes the orthogonal projection onto $F\in G_{n,k}$. We establish the validity of the above for a broad class of random polytopes in $\mathbb{R}^n$, that includes the case of random convex hulls with vertices chosen independently and uniformly from the interior or the surface of a convex body. We also discuss the case of unconditional convex bodies. Based on joint work with Nikos Skarmogiannis.

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