New Geometric Constants in Banach Spaces Related to the Inscribed Equilateral Triangles of Unit Balls

Yuankang Fu,Qi Liu,Yongjin Li

doi:10.3390/sym13060951

Abstract

Geometric constant is one of the important tools to study geometric properties of Banach spaces. In this paper, we will introduce two new geometric constants JL(X) and YJ(X) in Banach spaces, which are symmetric and related to the side lengths of inscribed equilateral triangles of unit balls. The upper and lower bounds of JL(X) and YJ(X) as well as the values of JL(X) and YJ(X) for Hilbert spaces and some common Banach spaces will be calculated. In addition, some inequalities for JL(X), YJ(X) and some significant geometric constants will be presented. Furthermore, the sufficient conditions for uniformly non-square and normal structure, and the necessary conditions for uniformly non-square and uniformly convex will be established.

Highlights

The study of geometric constants can be traced back to the concept of the modulus of convexity introduced by Clarkson [1] in the study of uniformly convex spaces
The modulus of convexity introduced by Clarkson [1] can be used to characterize uniformly convex spaces ([2], Lemma 2), the modulus of smoothness proposed by Day [3] can be used to characterize uniformly smooth spaces ([4], Theorem 2.5), the von Neumann–Jordan constant proposed by Clarkson [5] and the James constant proposed by Gao and Lau [6] can be used to characterize uniformly non-square spaces ([7], Theorem 2 and [8], Proposition 1)
We will introduce the following geometric constants related to the side lengths of the inscribed equilateral triangles of unit balls to study the geometric properties of Banach spaces

Summary

Introduction

The study of geometric constants can be traced back to the concept of the modulus of convexity introduced by Clarkson [1] in the study of uniformly convex spaces. In 2008, Alonso and Llorens-Fuster defined two geometric constants for Banach spaces by using the geometric means of the variable lengths of the sides of triangles with vertices x, − x and y, where x, y are points on the unit sphere of a normed space These constants are closely related to the modulus of convexity of some spaces, and they seem to represent a useful tool to estimate the exact values of the James and von Neumann–Jordan constants of some Banach spaces. Motivated by the fact that the problem about circles and their inscribed triangles is an important research topic of the Euclidean geometry and the works of the above two articles, we will introduce two new geometric constants JL ( X ) and YJ ( X ), in this paper, which are symmetric and related to the side lengths of inscribed equilateral triangles of unit balls in Banach spaces. The sufficient conditions for uniformly non-square and normal structure, and the necessary conditions for uniformly non-square and uniformly convex will be given by these two geometric constants JL ( X ) and YJ ( X )

Notations and Preliminaries

Some Examples

Conclusions