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Old and New on Strongly Subadditive/Superadditive Functions

Constantin P. Niculescu

TL;DR

This work broadens the theory of strong subadditivity and strong superadditivity by placing it in the setting of cones in ordered Banach spaces and extending characterizations to multivariate scenarios via second-order differences. It connects these notions to submodularity, leverages complete monotonicity through Bernstein-Hausdorff-Widder-Choquet representations to build strongly superadditive functions, and derives Popoviciu-type inequalities from weak majorization. Concrete instances include determinant and trace functionals, entropy-related inequalities, and a broadened toolbox for applications in analysis, probability, optimization, and quantum information. The Appendix consolidates the necessary background on ordered Banach spaces to support the framework.

Abstract

In this paper we provide insight into the classes of strongly subadditive/superadditive functions by highlighting numerous new examples and new results.

Old and New on Strongly Subadditive/Superadditive Functions

TL;DR

This work broadens the theory of strong subadditivity and strong superadditivity by placing it in the setting of cones in ordered Banach spaces and extending characterizations to multivariate scenarios via second-order differences. It connects these notions to submodularity, leverages complete monotonicity through Bernstein-Hausdorff-Widder-Choquet representations to build strongly superadditive functions, and derives Popoviciu-type inequalities from weak majorization. Concrete instances include determinant and trace functionals, entropy-related inequalities, and a broadened toolbox for applications in analysis, probability, optimization, and quantum information. The Appendix consolidates the necessary background on ordered Banach spaces to support the framework.

Abstract

In this paper we provide insight into the classes of strongly subadditive/superadditive functions by highlighting numerous new examples and new results.
Paper Structure (6 sections, 18 theorems, 90 equations)

This paper contains 6 sections, 18 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. The strong subadditivity/superadditivity in the context of several variables
  3. The connection with submodular functions
  4. Strong superadditivity from complete monotonicity
  5. An application to functional inequalities
  6. Appendix. Preliminaries on ordered Banach spaces

Key Result

Lemma 1

A continuous function $f:\mathbb{R}_{+}\rightarrow\mathbb{R}$ is convex if and only if it verifies the inequality

Theorems & Definitions (33)

  • Lemma 1
  • proof
  • Theorem 1
  • Proposition 1
  • Remark 1
  • Theorem 2
  • Theorem 3
  • Corollary 1
  • proof
  • Corollary 2
  • ...and 23 more