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

James Melbourne, Tomasz Tkocz, Katarzyna Wyczesany

TL;DR

This work extends multivariate anti-concentration bounds to an entropic setting by deriving pointwise lower bounds for densities of sums of independent vectors uniform on centered balls and connecting these bounds to noncentral section volumes of isotropic convex bodies. It introduces a subadditivity framework for Rényi entropies, enabling a multivariate extension of Bobkov–Chistyakov-type inequalities and providing explicit constants for the resulting concentration bounds. A key technical contribution is a probabilistic formula for the density of sums of uniforms, together with sharp density lower bounds via Archimedes-type projections and KR bounds. The paper then leverages a localization principle for even log-concave densities to obtain sharp lower bounds on noncentral sections, along with corollaries for isotropic convex bodies and discussions of sharpness. Finally, Rényi-entropy-based anti-concentration is established, and reversals under log-concavity are discussed in light of recent slicing-conjecture results, yielding two-sided bounds that tie together entropy, concentration, and geometric inequalities.

Abstract

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.

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

TL;DR

This work extends multivariate anti-concentration bounds to an entropic setting by deriving pointwise lower bounds for densities of sums of independent vectors uniform on centered balls and connecting these bounds to noncentral section volumes of isotropic convex bodies. It introduces a subadditivity framework for Rényi entropies, enabling a multivariate extension of Bobkov–Chistyakov-type inequalities and providing explicit constants for the resulting concentration bounds. A key technical contribution is a probabilistic formula for the density of sums of uniforms, together with sharp density lower bounds via Archimedes-type projections and KR bounds. The paper then leverages a localization principle for even log-concave densities to obtain sharp lower bounds on noncentral sections, along with corollaries for isotropic convex bodies and discussions of sharpness. Finally, Rényi-entropy-based anti-concentration is established, and reversals under log-concavity are discussed in light of recent slicing-conjecture results, yielding two-sided bounds that tie together entropy, concentration, and geometric inequalities.

Abstract

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.
Paper Structure (12 sections, 12 theorems, 77 equations)

This paper contains 12 sections, 12 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Noncentral sections
  3. Subadditivity of Rényi entropy
  4. Sums of uniforms: Proof of Theorem \ref{['thm:low-bd']}
  5. A probabilistic formula
  6. Probabilistic bounds
  7. Noncentral sections on effective support: Proofs of Theorem \ref{['thm:low-bd-at-sqrt3']} and Corollary \ref{['cor:noncent']}
  8. Proof of Theorem \ref{['thm:low-bd-at-sqrt3']}
  9. Proof of Corollary \ref{['cor:noncent']}
  10. Proof of Remark \ref{['rem:noncent-sharp']}
  11. Rényi entropy: Proof of Theorem \ref{['thm:Renyi-anti-conc']}
  12. Reversals under log-concavity

Key Result

Theorem 1

Let $d \geq 1$. Let $U_1, U_2, \dots$ be i.i.d. random vectors uniform on the unit Euclidean ball $B_2^d$ in $\mathbb{R}^d$. There is a positive constant $c_d$ depending only on $d$ such that for every $n \geq 1$ and real numbers $a_1, \dots, a_n$ with $\sum_{j=1}^n a_j^2 = 1$, we have where $p$ is the density of the random vector $\,\sum_{j=1}^n a_jU_j$.

Theorems & Definitions (28)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • proof
  • Corollary 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Theorem 8
  • Corollary 9
  • ...and 18 more