Diameters of sections and coverings of convex bodies

Abstract We study the diameters of sections of convex bodies in R N determined by a random N × n matrix Γ , either as kernels of Γ * or as images of Γ . Entries of Γ are independent random variables satisfying some boundedness conditions, and typical examples are matrices with Gaussian or Bernoulli random variables. We show that if a symmetric convex body K in R N has one well bounded k-codimensional section, then for any m > ck random sections of K of codimension m are also well bounded, where c ⩾ 1 is an absolute constant. It is noteworthy that in the Gaussian case, when Γ determines randomness in sense of the Haar measure on the Grassmann manifold, we can take c = 1 .

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