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 $st_A$ 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.