The central limit problem for convex bodies

Abstract

Abstract. It is shown that every symmetric convex body which satisfies a kind of weak law of large numbers has the property that almost all its marginal distributions are approximately Gaussian. Several quite broad classes of bodies are shown to satisfy the condition.

Volume distribution in convex bodies

“…Our idea is to put K in another position, namely Löwner's minimal diameter position, in which we show in Proposition 4.10 that the diameter is not larger than C λ n 1−λ , where λ depends only on α, the 2-convexity constant of K and C > 0 is a universal constant. We conclude Proposition 1.5 by proving a version of a Theorem from [ABP03] about the existence of Gaussian marginals, where the assumption of being in isotropic position is removed (see Theorem 5.3). Further developments on the existence of Gaussian marginals of uniformly convex bodies are discussed in [Mil06b].…”

“…The motivation for this requirement comes from [ABP03], where it was shown that if an isotropic 2-convex body has small-diameter in the above sense, then most of its marginals are approximately Gaussian (see [ABP03] or Section 5 for more details). It is easy to check that this requirement is indeed satisfied by all the l n p unit balls for 1 < p ≤ 2 (normalized to have the appropriate volume).…”

“…Theorem 5.3 (Generalization of [ABP03]). Let K be a centrallysymmetric convex body in R n of volume 1, and assume that for some ρ > 0 and ǫ < 1/2:…”

“…Let X denote a uniformly distributed vector inside a convex set K ⊂ R n of volume one. In its weakest form, a conjecture of Antilla, Ball and Perissinaki [ABP03], states that for some non-zero vector θ ∈ R n , the random variable X, θ is very close to a Gaussian random variable. That is, the total variation distance between the random variable X, θ and a corresponding Gaussian random variable, is smaller than ε n , where ε n is a sequence tending to zero, that depends solely on n. In this note, we verify the following (see Theorem 5.5 for an exact formulation): Proposition 1.5.…”

On Gaussian Marginals of Uniformly Convex Bodies

“…Theorem 1.1 (Generalized from [ABP03]). Assume that (1.1) holds for a centrally-symmetric convex body K. Then for any 0 < δ < c:…”