Pairs of subsets of spheres and Cartesian products thereof with the same distribution of distance

TL;DR

This paper investigates the distribution of distances within subsets of spheres and their Cartesian products, establishing conditions under which these distributions are equal or depend solely on measure differences.

Contribution

It proves new relationships between measure density functions of distance for subsets of spheres and their products, revealing invariance properties and conditions for equal distributions.

Findings

Distance distributions depend only on measure differences for partitions of spheres.

Equal measure subsets of spheres have identical distance distributions.

Distribution equality extends to Cartesian products of spheres with inherited metrics.

Abstract

We prove the following three statements: 1) Let $(A, \bar A)$ be a partition of the spherical surface $S^n$ into two measurable sets. Let $st_A$ and $st_{\bar A}$ be their measure density functions of distance. Then $|st_A - st_{\bar A}|$ depends only on the difference of their $n$-areas. 2) If the spherical surface $S^n$ is divided in two measurable subsets $A$ and $\bar A$ of equal $n$-surface, then these two subsets have the same distribution of distance. 3) Let there be a pair $(S, S')$ of subsets of a sphere $S^{n}$ such that $st_S = st_{S'}$. Then their complementary subsets satisfy $st_{\bar S} = st_{\bar S'}$ and $st_{S, \bar S} = st_{S', \bar S'}$, where $st_{A, B}$ is the measure density function of distance between a point in $A$ and a point in $B$. Furthermore, it is shown that the statements remain true when $S^n$ is substituted by the Cartesian product $S^{n_1} \times ... \times S^{n_r}$ endowed with the metric which is naturally inherited from its factors.