A Berry-Esseen type inequality for convex bodies with an unconditional basis

By Bo'az Klartag. 2007

126 citations recorded by the source corpus.

Suppose X = (X1, . . . , Xn) is a random vector, distributed uniformly in a convex body $${K \subset \mathbb R^n}$$ . We assume the normalization $${\mathbb E X_i^2 = 1}$$ for i = 1, . . . , n. The body K is further required to be invariant under coordinate reflections, that is, we assume that (±X1, . . . , ±Xn) has the same distribution as (X1, . . . , Xn) for any choice of signs. Then, we show that$$ \mathbb E \left( \, |X| - \sqrt{n} \, \right)^2 \leq C^2,$$where C ≤ 4 is a positive universal constant, and | · | is the standard Euclidean norm in $${\mathbb R^n}$$ . The estimate is tight, up to the value of the constant. It leads to a Berry-Esseen type bound in the central limit theorem for unconditional convex bodies.

Subfields: Applied Mathematics, Molecular Biology, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/4bdcda8939eddd32801401c3cfe9c02eed9357ea

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Citations in the corpus

Bourgain’s slicing problem and KLS isoperimetry up to polylog (2022, methodology): “(15) The following inequality was proven in the case of the uniform measure on a convex body in [32] and for a general log-concave measure in Barthe and Klartag [4].”
Logarithmic bounds for isoperimetry and slices of convex sets (2023, unknown): “See [4, 29] for basic properties of the H(μ)-norm and for the inequality λ1(μ) · ‖f‖(2)H−1(μ) ≤ ‖f‖(2)L2(μ).”
Brascamp–Lieb-Type Inequalities on Weighted Riemannian Manifolds with Boundary (2013, background): “We mention in passing another celebrated application of the latter duality argument in the Complex setting, namely Hörmander’s L2 estimate [13], but we refrain from attempting to generalize it here; further more recent applications may be found in [3,12,15,16,20].”
Cone-volume measure of general centered convex bodies (2016, unknown): “Klartag [26].”
Local $L^p$-Brunn-Minkowski inequalities for $p<1$ (2017, background): “[30, 3, 28, 32, 33]), one may obtain an improved estimate for p above.”
Local Lp-Brunn-Minkowski inequalities for p<1 (2017, background): “ ≥ (1 −λ)V (K0) p n + λV(K1) p n n p ∀λ ∈ [0,1]. 6.3 Examples Of course, using the known estimates on CPoin(K) which improve over the general (6.5) for various classes of convex bodies K (see e.g. [34, 3, 32, 37, 38]), one may obtain an improved estimate for p above. It will be instructive in this ”
On Stein's method for multivariate normal approximation (2009, unknown): “, [19],[20],[15].”
Poincaré and Brunn-Minkowski inequalities on the boundary of weighted Riemannian manifolds (2017, methodology): “Further applications of L2-duality in the context of obtaining Poincaré-type inequalities may be found in [28, 27, 40, 33, 7, 34].”