Entropic versions of Bergström's and Bonnesen's inequalities

TL;DR

The paper develops entropic analogues of Bergström's determinant inequality and Bonnesen's volume inequality by relating entropy power and Fisher information to classical geometric inequalities. It introduces a conditional entropy framework and proves an entropic Bergström-type inequality (Theorem B2) and an entropic Bonnesen refinement, with a complete equality characterization in the Gaussian case, showing these results strengthen the EPI in dimensions $d>1$ and reduce to Bergström's result in $d=1$. A companion conditional/Fisher-information inequality is derived, using a Blachman–Stam approach on transformed variables and extending to directional forms via $I_{P^u}$, which yields a stronger bound than the standard Fisher information inequality. The work also provides corollaries for sections and isoperimetric-type entropy inequalities, and situates these entropic results within the broader entropy–volume analogy and information-theoretic proofs. Overall, it expands the toolkit for entropic inequalities with precise equality conditions and connections to Gaussian structure.

Abstract

We establish analogues of the Bergström and Bonnesen inequalities, related to determinants and volumes respectively, for the entropy power and for the Fisher information. The obtained inequalities strengthen the well-known convolution inequality for the Fisher information as well as the entropy power inequality in dimensions $d>1$, while they reduce to the former in $d=1$. Our results recover the original Bergström inequality and generalize a proof of Bergström's inequality given by Dembo, Cover and Thomas. We characterize the equality case in our entropic Bonnesen inequality.

Theorems & Definitions (18)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 5
• proof
• Theorem 6: Entropy analogue of Bergström's inequality
• proof
• Remark 7
• Corollary 8: Entropy analogue of Bonnesen's inequality