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Inequalities Revisited

Raymond W. Yeung

TL;DR

The paper establishes a geometrical framework for universally quantified inequalities rooted in entropy, and uses it to reinterpret classic inequalities such as the AM–GM, Markov, and Cauchy–Schwarz as outer-bound characterizations of achievable regions. It then extends the formalism to entropy inequalities, distinguishing Shannon-type from non-Shannon-type inequalities, and surveys the deep connections to network coding, probability, group theory, matrix theory, Kolmogorov complexity, and quantum mechanics. Key insights include complete characterizations for AM–GM, almost-complete descriptions for Markov, and sharp bounds in Cauchy–Schwarz depending on dimension, along with a comprehensive view of entropy regions ${\Gamma}_n^*$ and their constrained counterparts. The work suggests a unified method to uncover new inequalities across mathematics by translating problems into geometric regions and their outer bounds, with practical implications for information theory, coding, and related fields.

Abstract

In the past over two decades, very fruitful results have been obtained in information theory in the study of the Shannon entropy. This study has led to the discovery of a new class of constraints on the Shannon entropy called non-Shannon-type inequalities. Intimate connections between the Shannon entropy and different branches of mathematics including group theory, combinatorics, Kolmogorov complexity, probability, matrix theory, etc, have been established. All these discoveries were based on a formality introduced for constraints on the Shannon entropy, which suggested the possible existence of constraints that were not previously known. We assert that the same formality can be applied to inequalities beyond information theory. To illustrate the ideas, we revisit through the lens of this formality three fundamental inequalities in mathematics: the AM-GM inequality in algebra, Markov's inequality in probability theory, and the Cauchy-Scharwz inequality for inner product spaces. Applications of this formality have the potential of leading to the discovery of new inequalities and constraints in different branches of mathematics.

Inequalities Revisited

TL;DR

The paper establishes a geometrical framework for universally quantified inequalities rooted in entropy, and uses it to reinterpret classic inequalities such as the AM–GM, Markov, and Cauchy–Schwarz as outer-bound characterizations of achievable regions. It then extends the formalism to entropy inequalities, distinguishing Shannon-type from non-Shannon-type inequalities, and surveys the deep connections to network coding, probability, group theory, matrix theory, Kolmogorov complexity, and quantum mechanics. Key insights include complete characterizations for AM–GM, almost-complete descriptions for Markov, and sharp bounds in Cauchy–Schwarz depending on dimension, along with a comprehensive view of entropy regions and their constrained counterparts. The work suggests a unified method to uncover new inequalities across mathematics by translating problems into geometric regions and their outer bounds, with practical implications for information theory, coding, and related fields.

Abstract

In the past over two decades, very fruitful results have been obtained in information theory in the study of the Shannon entropy. This study has led to the discovery of a new class of constraints on the Shannon entropy called non-Shannon-type inequalities. Intimate connections between the Shannon entropy and different branches of mathematics including group theory, combinatorics, Kolmogorov complexity, probability, matrix theory, etc, have been established. All these discoveries were based on a formality introduced for constraints on the Shannon entropy, which suggested the possible existence of constraints that were not previously known. We assert that the same formality can be applied to inequalities beyond information theory. To illustrate the ideas, we revisit through the lens of this formality three fundamental inequalities in mathematics: the AM-GM inequality in algebra, Markov's inequality in probability theory, and the Cauchy-Scharwz inequality for inner product spaces. Applications of this formality have the potential of leading to the discovery of new inequalities and constraints in different branches of mathematics.

Paper Structure

This paper contains 16 sections, 11 theorems, 120 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The AM-GM Inequality
  3. Markov's Inequality
  4. Cauchy-Schwarz Inequality
  5. Entropy Inequalities
  6. Unconstrained and Constrained Entropy Inequalities
  7. Shannon-Type Inequalities
  8. Beyond Shannon-Type Inequalities
  9. Connections with Other Fields
  10. Network Coding
  11. Probability Theory
  12. Group Theory
  13. Matrix Theory
  14. Kolmogorov Complexity
  15. Quantum Mechanics
  16. ...and 1 more sections

Key Result

Proposition 1

For any $(a,g) \in \Upsilon$, there exist $x,y \ge 0$ such that i.e., $a$ and $g$ are the AM and GM of the list of nonnegative numbers $x, y$, respectively.

Figures (5)

  • Figure 1: The region $\Upsilon$ in $\mathbb{R}^2$ and an inequality $f(a,g) \ge 0$ implied by $g \ge 0$ and $a \ge g$.
  • Figure 2: An illustration of the region $R_f$ for $f = 2a - g$.
  • Figure 3: The region $\Psi_c$ in $\mathbb{R}^2$.
  • Figure 4: An illustration of the non-Shannon-type inequality ZY98.
  • Figure 5: The butterfly network that illustrates the advantage of network coding.

Theorems & Definitions (13)

  • Proposition 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Corollary 1
  • Example 1
  • Definition 1
  • Theorem 5
  • Theorem 6
  • ...and 3 more