Randomizing properties of convex high-dimensional bodies and some geometric inequalities

Properties of convex bodies related to uniform distribution are studied. In particular, a low bound for the norm of the sum of independent geometrically distributed vectors is obtained. It extends the previously studied case of identically distributed vectors by Bourgain, Meyer, Milman and Pajor and solves a problem raised there. Another corollary asserts that any finite dimensional normed space has a “random cotype 2”. To cite this article: E. Gluskin, V. Milman, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 875–879.