A Rényi entropy interpretation of anti-concentration and noncentral sections of convex bodies

We extend Bobkov and Chistyakov’s (2015) upper bounds on concentration functions of sums of independent random variables to a multivariate entropic setting. The approach is based on pointwise estimates on densities of sums of independent random vectors uniform on centred Euclidean balls. In this vein, we also obtain sharp bounds on volumes of noncentral sections of isotropic convex bodies.

💡 Research Summary

The paper “A Rényi entropy interpretation of anti‑concentration and non‑central sections of convex bodies” extends the classical anti‑concentration bounds for sums of independent random variables, originally due to Rogozin, Esseen, Kesten, and most recently Bobkov and Chistyakov (2015), into a multivariate information‑theoretic framework based on Rényi entropies. The authors’ main contributions can be grouped into four interrelated themes.

1. Multivariate anti‑concentration via Rényi entropy.
   The concentration function of a random vector (X\in\mathbb R^{d}) is defined as
   \