Inscribed triangles in the unit sphere and a new class of geometric constants

TL;DR

This paper introduces a new geometric constant GL(X) related to inscribed triangles in the unit sphere, explores its properties, and establishes its connections with existing constants, enriching the understanding of geometric constants in Banach spaces.

Contribution

The paper proposes a novel geometric constant GL(X), investigates its properties, and relates it to existing constants, expanding the theoretical framework of geometric analysis.

Findings

Introduction of the new constant GL(X) and its basic properties

Relations established between GL(X) and other known geometric constants

Characterization of generalized forms of GL(X)

Abstract

We will introduce a new geometric constant GL(X) based on the constant H(X) proposed by Gao. We first further survey the constant H(X) and discuss some of the properties of this constant that have not yet been discovered. Next, we focus on a new constant GL(X) along with some of its basic properties. In addition, we show some relations between the well-known geometric constants and GL(X) through some inequalities. Finally, we characterize some generalized forms of the constant GL(X).