Generalization of Mrs. Gerber's Lemma

TL;DR

This paper generalizes Mrs. Gerber's Lemma by establishing broader convexity conditions for entropy functions, simplifying proofs, and enhancing applications in binary broadcast channels.

Contribution

It proves a generalized convexity condition for entropy functions, extending Mrs. Gerber's Lemma and simplifying its proof and applications.

Findings

Generalized convexity condition for entropy functions.

Simplified proof of Mrs. Gerber's Lemma.

Enhanced application in binary broadcast channels.

Abstract

Mrs. Gerber's Lemma (MGL) hinges on the convexity of $H(p*H^{-1}(u))$, where $H(u)$ is the binary entropy function. In this work, we prove that $H(p*f(u))$ is convex in $u$ for every $p\in [0,1]$ provided $H(f(u))$ is convex in $u$, where $f(u) : (a, b) \to [0, \frac12]$. Moreover, our result subsumes MGL and simplifies the original proof. We show that the generalized MGL can be applied in binary broadcast channel to simplify some discussion.