A Generalized Central Limit Conjecture for Convex Bodies
Lecture Notes in Mathematics(2020)
摘要
The central limit theorem for convex bodies says that with high probability the marginal of an isotropic log-concave distribution along a random direction is close to a Gaussian, with the quantitative difference determined asymptotically by the Cheeger/Poincare/KLS constant. Here we propose a generalized CLT for marginals along random directions drawn from any isotropic log-concave distribution; namely, for x, y drawn independently from isotropic log-concave densities p, q, the random variable < x, y > is close to Gaussian. Our main result is that this generalized CLT is quantitatively equivalent (up to a small factor) to the KLS conjecture. Any polynomial improvement in the current KLS bound of n(1/4) in R-n implies the generalized CLT, and vice versa. This tight connection suggests that the generalized CLT might provide insight into basic open questions in asymptotic convex geometry.
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关键词
generalized central limit conjecture
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