On a multi-integral norm defined by weighted sums of log-concave random vectors

Let C C and K"> K encoding="application/x-tex">K be centrally symmetric convex bodies in alttext="double-struck upper R Superscript n"> R n encoding="application/x-tex">{\mathbb R}^n . We show that if is isotropic then ‖ mathvariant="bold">t s , = ∫ σ<!-- σ encoding="application/x-tex">M(K)≔\int _{S^{n-1}}\|\xi \|_Kd\sigma (\xi ) This reduces a question V. Milman to problem estimating from above parameter encoding="application/x-tex">M(K) an body. The proof based on observation combines results Eldan, Lehec Klartag slicing problem: If alttext="mu"> μ<!-- μ encoding="application/x-tex">\mu log-concave probability measure then, any body we have I mu 2 period"> I . I_1(\mu ,K)≔\int _{{\mathbb R}^n}\|x\|_K\,d\mu (x)\leqslant c_2\sqrt {n}(\log n)^5\,M(K). illustrate use this inequality with further applications.