The Spherical Ratio of Two Points and its Integral Properties

TL;DR

This paper explores the spherical ratio of two points in multi-dimensional space, establishing integral properties, a new integration technique, and a One Radius Theorem, extending classical planar results to higher dimensions.

Contribution

It introduces a new integral technique and proves a One Radius Theorem for the spherical ratio in multi-dimensional spaces, extending classical planar geometric interpretations.

Findings

Derived integral properties of the spherical ratio.

Established a new integration technique involving segment ratios.

Proved a One Radius Theorem for the spherical ratio.

Abstract

At the end of 19-th century, 1874, Hermann Schwartz found that for every point inside a planar disk, a two-dimensional Poisson Kernel can be written as a ratio of two segments, which he called as the geometric interpretation of that Kernel \cite{Schwarz}. About 90 years later, Lars Ahlfors, in his textbook, called this ratio as interesting \cite{Ahlfors}. We shall see here that every two different points, in a multidimensional space also define a similar ratio of two segments. The main goal of this paper is to study some integral properties of the ratio as well as introduce and proof One Radius Theorem for Spherical Ratio of two points. In particular, we introduce a new integration technique involving the ratio of two segments and a non-trivial integral equivalence leading to an integral relationship between Newtonian Potential and Poisson Kernel in any multi-dimensional space.