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

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 $s{t}_A$ and $s{t}_{\bar{A}}$ be their measure density functions of distance. Then $|s{t}_A-s{t}_{\bar{A}}|$ 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 $(S,S^{\prime })$ of subsets of a sphere $S^n$ such that $s{t}_S=s{t}_{S^{\prime }}$. Then their complementary subsets satisfy $s{t}_{\bar{S}}=s{t}_{{\bar{S}}^{\prime }}$ and $s{t}_{S,\bar{S}}=s{t}_{S^{\prime },{\bar{S}}^{\prime }}$, where $s{t}_{A,B}$ 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.

arXiv comment1 figure