Sharp Min-Entropy Power Inequalities

Definitions and mathematical statement

Let $d$ and $N$ be positive integers, and let $X_1,\ldots,X_N$ be independent random vectors in ${\mathbb{R}}^d$. The supplied cleaned statement asks for the sharp constant and all extremizers in a Rényi entropy-power inequality for their sum, for orders other than $0,1,\infty$. However, the record does not specify the inequality, its admissible distribution class, or its full range of orders.

To describe a conventional interpretation, suppose that a random vector $X$ has a Lebesgue density $f$. For $0<\alpha<\infty$, $\alpha\ne1$, and $0<\int_{{\mathbb{R}}^d}f(x)^\alpha\,{\,\mathrm{d}} x<\infty$, define \[ h_\alpha(X)=\frac{1}{1-\alpha}\log\!\left(\int_{{\mathbb{R}}^d}f(x)^\alpha\,{\,\mathrm{d}} x\right), \qquad P_\alpha(X)=\exp\!\left(\frac{2h_\alpha(X)}{d}\right). \] These definitions use the natural logarithm. If the intended inequality has the conventional linear form, its target would be to determine the largest constant $C_{\alpha,d,N}$ such that \[ P_\alpha(X_1+\cdots+X_N) \ge C_{\alpha,d,N}\sum_{i=1}^{N}P_\alpha(X_i) \] for every admissible independent $N$-tuple, and to characterize every admissible tuple attaining equality. The displayed inequality is a conditional reconstruction: neither this exact form nor the density and finiteness hypotheses are explicitly supplied by the source record. Consequently, an unconditional extremal problem cannot be specified from the record alone. Whether orders below zero, infinite entropy powers, or distributions without densities are intended remains unresolved. For $N=1$, the displayed reconstruction is tautologically sharp with constant $1$ when the entropy power is positive and finite; the source does not state its intended lower bound on $N$.

The original excerpt concerns a different order. For a density $f$ with finite positive essential supremum, the conventional min-entropy and its power are \[ h_\infty(X)=-\log\|f\|_{L^\infty({\mathbb{R}}^d)}, \qquad P_\infty(X)=\|f\|_{L^\infty({\mathbb{R}}^d)}^{-2/d}. \] The excerpt reports a min-entropy power inequality with a constant independent of $N$ and sharp for each fixed dimension $d$. It supplies neither the inequality nor its constant or equality cases, and it does not itself state the broader finite-order problem above. The supplied background additionally describes a sharper dimension-and-summand-dependent constant as an open target. Sharpness uniformly over $N$ and sharpness for each fixed $N$ are distinct requirements; the record does not specify their relationship precisely. These unresolved qualifications require mathematical and source review.

Short English statement

The cleaned record asks for optimal constants and all equality cases for Rényi entropy-power inequalities for sums of independent vectors in ${\mathbb{R}}^d$, excluding orders $0,1,\infty$. Its original excerpt instead reports a min-entropy inequality. The exact inequality, admissible distributions, and order range must be confirmed before the conditional formulation above can serve as a precise problem statement.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005065$, title Sharp Min-Entropy Power Inequalities, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied status is open; its statement status is corrected_verified. These are registry labels, not independent verification by this draft.
• Supplied report citation: New Mathematical Techniques in Information Theory (2023), Oberwolfach Reports, DOI 10.4171/owr/2022/14, pp. 683--707. Literature-source label: Oberwolfach Report 14/2022. https://doi.org/10.4171/owr/2022/14. The record assigns proposed year 2022; the citation's 2023 year and the report's 2022 designation are retained without resolving their bibliographic relationship.
• Attribution in the original excerpt: Min-Entropy Inequalities on Algebraic Structures, Mokshay Madiman, joint work with James Melbourne, Cyril Roberto, and Peng Xu. No separate proposer is identified in the supplied record.
• Supplied literature source: Xu--Melbourne--Madiman, min-entropy power inequality for groups. https://doi.org/10.1109/ISIT.2017.8006613.
• Supplied literature assessment, checked on 2026-08-22: the universal-constant min-entropy inequality is described as established, while the sharper dimension-and-summand-dependent constant and broader Rényi entropy-power extremizer questions are described as open. This is supplied dated curation derived from the OWR record, not independently verified literature assessment. The record also claims that its cleaned statement was checked against the official report; this draft does not independently confirm that claim.