Geometric Constants in Banach Spaces Related to the Inscribed Quadrilateral of Unit Balls

Asif Ahmad,Yongjin Li,Qi Liu

doi:10.3390/sym13071294

Abstract

We introduce a new geometric constant Jin(X) based on a generalization of the parallelogram law, which is symmetric and related to the length of the inscribed quadrilateral side of the unit ball. We first investigate some basic properties of this new coefficient. Next, it is shown that, for a Banach space, Jin(X) becomes 16 if and only if the norm is induced by an inner product. Moreover, its properties and some relations between other well-known geometric constants are studied. Finally, a sufficient condition which implies normal structure is presented.

Highlights

It is worth noting that the background meaning of many famous geometric constants is closely related to the description of inner product spaces
Fu et al introduced the constants: JL ( X ) = inf{k x1 − x2 k, k x2 − x3 k, k x1 − x3 k : k x1 k = k x2 k = k x3 k = 1, x1 + x2 + x3 = 0}; YJ ( X ) = sup{k x1 − x2 k, k x2 − x3 k, k x1 − x3 k : k x1 k = k x2 k = k x3 k = 1, x1 + x2 + x3 = 0}, which are related to the side lengths of the inscribed triangles of unit balls to study the geometric properties of Banach spaces
A Banach space X is said to have weak normal structure if each weakly compact convex set K in X that contains more than one point has normal structure

Summary

Introduction

It is worth mentioning that geometric constants play a vital role as a tool for solving other problems, such as in the study of Banach–Stone theorem, Bishop–Phelps–Bollobás theorem, and Tingley’s Problem. These are important research topics in the area of functional analysis and we recommend that readers refer to the literature [7,8,9]. It is worth noting that the background meaning of many famous geometric constants is closely related to the description of inner product spaces.

Preliminaries

The Coefficient of Weak Orthogonality

Conclusions