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On the problem of reversibility of the entropy power inequality

Sergey G. Bobkov, Mokshay M. Madiman

TL;DR

This work investigates the reversibility of the entropy power inequality (EPI) under convexity constraints. While reversibility can occur for certain summand distributions with strong concavity after placing them in a special affine position, the authors prove that no universal reversal bound exists over the entire class of convex distributions, using a 1D counterexample based on a truncation of Pareto densities. They introduce a difference-measure inequality for convex measures and derive entropy bounds for sums and differences via a Rogers–Shephard–type approach, highlighting sharp contrasts between log-concave and fully convex regimes. The paper also discusses implications for entropic distance to normality and situates the results within the broader inverse Brunn–Minkowski framework, clarifying the limits of reversibility beyond log-concave settings.

Abstract

As was shown recently by the authors, the entropy power inequality can be reversed for independent summands with sufficiently concave densities, when the distributions of the summands are put in a special position. In this note it is proved that reversibility is impossible over the whole class of convex probability distributions. Related phenomena for identically distributed summands are also discussed.

On the problem of reversibility of the entropy power inequality

TL;DR

This work investigates the reversibility of the entropy power inequality (EPI) under convexity constraints. While reversibility can occur for certain summand distributions with strong concavity after placing them in a special affine position, the authors prove that no universal reversal bound exists over the entire class of convex distributions, using a 1D counterexample based on a truncation of Pareto densities. They introduce a difference-measure inequality for convex measures and derive entropy bounds for sums and differences via a Rogers–Shephard–type approach, highlighting sharp contrasts between log-concave and fully convex regimes. The paper also discusses implications for entropic distance to normality and situates the results within the broader inverse Brunn–Minkowski framework, clarifying the limits of reversibility beyond log-concave settings.

Abstract

As was shown recently by the authors, the entropy power inequality can be reversed for independent summands with sufficiently concave densities, when the distributions of the summands are put in a special position. In this note it is proved that reversibility is impossible over the whole class of convex probability distributions. Related phenomena for identically distributed summands are also discussed.

Paper Structure

This paper contains 4 sections, 6 theorems, 58 equations.

Table of Contents

  1. The reversibility problem for the entropy power inequality
  2. A "difference measure" inequality for convex measures
  3. Proof of Theorem \ref{['bm-thm:neg']}
  4. Remarks

Key Result

Theorem 1.1

Let $X$ and $Y$ be independent random vectors in $\mathbb{R}^n$ with densities of the form bm-cvx with $\beta \geq 2n+1$, $\beta \geq \beta_0 n$$(\beta_0 > 2)$. There exist linear volume preserving maps $T_i:\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that where $\widetilde{X} = T_1(X)$, $\widetilde{Y} = T_2(Y)$, and where $C_{\beta_0}$ is a constant, depending on $\beta_0$, only.

Theorems & Definitions (6)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Corollary 2.3