High dimensional random sections of isotropic convex bodies
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We study two properties of random high dimensional sections of convex bodies. In the first part of the paper we estimate the central section function | K ∩ F⊥ |n - k1 / k for random F ∈ Gn, k and K ⊂ Rn a centrally symmetric isotropic convex body. This partially answers a question raised by V.D. Milman and A. Pajor (see [V.D. Milman, A. Pajor, Isotropic positions and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space, in: Lecture Notes in Math., vol. 1376, Springer, 1989, p. 88]). In the second part we show that every symmetric convex body has random high dimensional sections F ∈ Gn, k with outer volume ratio bounded byovr (K ∩ F) ≤ C frac(n, n - k) log (1 + frac(n, n - k)) . © 2009 Elsevier Inc. All rights reserved.
author list (cited authors)
Alonso-Gutiérrez, D., Bastero, J., Bernués, J., & Paouris, G.