Gaussian Marginals of Probability Measures with Geometric Symmetries

Let K be a convex body in the Euclidean space Rn, n ≥ 2, equipped with its standard inner product 〈·, ·〉 and Euclidean norm | · |. Consider K as a probability space equipped with its uniform (normalized Lebesgue) measure μ. We are interested in k-dimensional marginals of μ, that is, the push-forward μ◦P−1 E of μ by the orthogonal projection PE onto a k-dimensional subspace E ⊂ Rn. The question of whether every convex body has 1-dimensional marginals which are close in some sense to a Gaussian measure is known as the central limit problem for convex bodies [1, 8]; a natural generalization is to ask the same question about k-dimensional marginals. Both of these questions were recently answered in the affirmative by Klartag [13], in the latter case for k ≪ logn log logn . Along the way partial results, involving different additional hypotheses and notions of closeness for probability measures, were proved by many authors in [1, 3, 4, 7, 8, 14, 15, 16, 19, 20] among others. Many of these papers treat marginals of more general probability measures μ than uniform measures on convex bodies. One common generalization is to log-concave measures, i.e., ones with a logarithmically concave density with respect to Lebesgue measure. Klartag’s recent results [13] are proved in this generality; some of the other results apply still more generally. In many of the papers mentioned above the existence of a (usually 1-dimensional) nearly Gaussian marginal μ ◦ P−1 E is proved probabilistically without identifying any concrete such subspace E. In [14], E. Meckes and the author proved Berry-Esseen theorems for specific 1-dimensional marginals when μ is invariant under certain types of geometric symmetries. The methods of [14] are insensitive to convexity and work for completely arbitrary probability measures satisfying the symmetry hypotheses. The main purpose of this paper is to prove versions of the results of [14] for k-dimensional marginals. We consider distributions which are 1-unconditional, meaning they are invariant under reflections in the coordinate hyperplanes of Rn (Theorem 2.2); distributions which are 1-symmetric, meaning they are 1-unconditional and also invariant under permutations of the coordinates (Theorem 2.1); or which possess all the symmetries of a centered regular simplex (Theorem 2.3). In [14] a symmetry hypothesis was introduced which simultaneously generalizes 1-unconditionality and the symmetries of a regular simplex. While the present methods would allow us to prove multivariate results under the same hypothesis, we have preferred for the sake of transparency to deal with