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Primal and dual characterizations of sign-symmetric norms

Nguyen Duy Cuong

Abstract

The paper studies primal and dual characterizations of a class of sign-symmetric norms on product vector spaces. Correspondences between these norms and a class of convex functions are established. Explicit formulas for the dual norm and the convex subdifferential of a given primal norm are derived. It is demonstrated that this class of norms is well-suited for studying properties and problems on product spaces. As an application, we study the von Neumann-Jordan constant of norms on product spaces and extend a classical result of Clarkson from Lebesgue spaces to general normed vector spaces.

Primal and dual characterizations of sign-symmetric norms

Abstract

The paper studies primal and dual characterizations of a class of sign-symmetric norms on product vector spaces. Correspondences between these norms and a class of convex functions are established. Explicit formulas for the dual norm and the convex subdifferential of a given primal norm are derived. It is demonstrated that this class of norms is well-suited for studying properties and problems on product spaces. As an application, we study the von Neumann-Jordan constant of norms on product spaces and extend a classical result of Clarkson from Lebesgue spaces to general normed vector spaces.
Paper Structure (9 sections, 22 theorems, 47 equations)

This paper contains 9 sections, 22 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Primal characterizations of sign-symmetric norms
  3. Dual norms
  4. Subdifferential characterzations
  5. Extremal principle
  6. The von Newmann-Jordan constant
  7. The $C_{\text{NJ}}$ for sign-symmetric norms
  8. A result of Clarkson revisited
  9. Conclusions

Key Result

Proposition 2.1

Let $\left\vert\left\vert\left\vert \cdot\right\vert\right\vert\right\vert\in \textbf{\rm N}_{X^n}$ and $x:=(x_1,\ldots,x_n)\in X^n$. Then

Theorems & Definitions (60)

  • Remark 2.1
  • Example 2.2
  • Proposition 2.1
  • proof
  • Remark 2.3
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Remark 2.4
  • ...and 50 more