Anttila, Milla; (2001) Concentration estimates and the central limit problem for convex bodies. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

It is shown that a large class of convex bodies satisfy a central limit property. In particular we show that if an isotropic, symmetric, convex body, K, has the property that most of its mass concentrates near its average radius, then its marginal distribution in direction theta, (whose density is given by scanning across K with hyperplanes perpendicular to theta), is close to a Gaussian in most directions. This closeness is shown in terms of the distribution function and the density function. We also show how the transportation method for obtaining concentration results works for the cube and, more importantly, we find the best constant possible using this method. This constant turns out to be better than those obtained by traditional methods and cannot be far from that which is best possible.

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