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

Suppose X = (X ₁, . . . , X _n ) 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 (±X ₁, . . . , ±X _n ) has the same distribution as (X ₁, . . . , X _n ) 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.