A Note on Homogeneous Functionsand Parallelogram Law

TL;DR

This paper presents a new characterization of the parallelogram law in normed linear spaces using homogeneous properties of positive definite functions, offering alternative conditions for this fundamental geometric property.

Contribution

It introduces a novel equivalence between the parallelogram law and homogeneous properties of positive definite functions in normed spaces.

Findings

Equivalent conditions to the parallelogram law are established.

Homogeneous positive definite functions characterize the parallelogram law.

Provides a new perspective on geometric properties in normed spaces.

Abstract

In this article, we investigate a new characterization of the parallelogram law in a normed linear space. We give equivalent conditions to the paralleogram law, in terms of the homogeneous property of a continuous positive definite function on the normed space.