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Generalized geometric constants related to Birkhoff orthogonality in Banach spaces

Junxiang Qi, Qian Li, Zhouping Yin, Qi Liu, Jiaye Bi, Yuankang Fu, Yongjin Li

Abstract

In this paper, based on Birkhoff orthogonality, we introduce two geometric constants $\boldsymbol{A}_{\boldsymbol{t}}^{\boldsymbol{B}}(\boldsymbol{X})$ and $\boldsymbol{D}_{\boldsymbol{t}}^{\boldsymbol{B}}(\boldsymbol{X})$ in Banach spaces, which generalize the skew geometric constants related to Birkhoff orthogonality. We systematically investigate the basic properties of the two constants, including their upper and lower bounds, and establish the equivalent characterizations for Banach spaces being uniformly non-square. Additionally, we explore the relationship between $\boldsymbol{D}_{\boldsymbol{t}}^{\boldsymbol{B}}(\boldsymbol{X})$ and the modulus of convexity $\boldsymbolδ_{\boldsymbol{X}}(\boldsymbol{\varepsilon})$. Finally, we explore several applications of the two newly proposed geometric constants.

Generalized geometric constants related to Birkhoff orthogonality in Banach spaces

Abstract

In this paper, based on Birkhoff orthogonality, we introduce two geometric constants and in Banach spaces, which generalize the skew geometric constants related to Birkhoff orthogonality. We systematically investigate the basic properties of the two constants, including their upper and lower bounds, and establish the equivalent characterizations for Banach spaces being uniformly non-square. Additionally, we explore the relationship between and the modulus of convexity . Finally, we explore several applications of the two newly proposed geometric constants.
Paper Structure (4 sections, 22 theorems, 107 equations, 3 figures)

This paper contains 4 sections, 22 theorems, 107 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The constant $A^B_{t}(X)$
  3. The constant $D_{t}^{B}(X)$
  4. Some applications of $A^B_{t}(X)$ and $D_{t}^{B}(X)$

Key Result

Lemma 1

32 (1) The function $f(t)=\|x+t y\|+\|t x-y\|$ is a convex function of $t$ on $\mathbb{R}$. (2) The function $g(t)=\|t x+y\|+\|x-t y\|$ is a convex function of $t$ on $\mathbb{R}$.

Figures (3)

  • Figure 1: Illustration of skew isosceles orthogonality
  • Figure 2: Geometric explanation of $\ell_\infty-\ell_1$.
  • Figure 3: The unit sphere of the affine regular hexagon

Theorems & Definitions (48)

  • Lemma 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Lemma 2
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • ...and 38 more