New geometric constants of isosceles orthogonal type

TL;DR

This paper introduces a new geometric constant based on isosceles orthogonality and the parallelogram law, exploring its properties, relation to non-square spaces, and connections to existing constants.

Contribution

It defines a novel orthogonal geometric constant, analyzes its properties, and establishes its relation to the inner product structure and other known constants.

Findings

The constant equals 1 if and only if the norm is induced by an inner product.

The constant has specific properties related to the non-square property of spaces.

The generalized constant extends the original and maintains key properties.

Abstract

Based on the parallelogram law and isosceles orthogonality, we define a new orthogonal geometric constant. We first discuss some basic properties of this new constant. Next, we consider the relation between the constant and the uniformly non-square property. Moreover, a generalized constant is also introduced and some basic properties are presented. It is shown that, for a normed space, the constant value is equal to 1 if and only if the norm can be induced by the inner product. Finally, we verify that this constant is closely related to the well-known geometric constants through some inequalities.