Some Geometric Constants Related to the Midline of Equilateral Triangles in Banach Spaces

Bingren Chen,Qi Liu,Yongjin Li,Zhijian Yang

doi:10.3390/sym14020348

Abstract

We will introduce some new geometric constants based on the constant H(X) proposed by Gao and the constant A2(X) proposed by M. Baronti et al. We first provide a study of a new constant M1(X) closely related to the midlines of equilateral triangles, including a discussion of some of its properties and the connections with other parameters of the sphere. Next, we focus on a new constant M2(X) and its generalized form M2(X,p,q), along with some of their basic properties. Finally, we concentrate on a new constant M3(X) and discuss some of its properties.

Highlights

It is interesting to study the non-symmetrical properties of these constants and hopefully obtain some significant results
Some Results Related to New Constants M2 ( X ) and M3 ( X )
In light of the asymmetric geometric constant H ( X ) proposed by Gao and the Baronti type constant A2 ( X ), we intend to introduce a new geometric constant M1 ( X ), which measures the lengths of two midlines of a triangle with one vertex at the center of the unit ball and the other on the unit sphere

Summary

Introduction

It is interesting to study the non-symmetrical properties of these constants and hopefully obtain some significant results. A Banach space X is called uniformly non-square if there exists δ ∈ (0, 1) such that k x −yk k x +yk for any x, y ∈ SX , we have either 2 ≤ 1 − δ or 2 ≤ 1 − δ. The James constant (Gao [7], Gao and Lau [8]): J ( X ) := sup{min(k x + yk, k x − yk) : k x k = kyk = 1}; (ii)

Results

Conclusion