Space Distance

The maximum distance in 3-D can occur no more than $2n-2$ times. Also, there exists a fixed number $c$ such that no distance determined by a set of $n$ points in 3-D space occurs more than $cn^{5/3}$ times. The maximum distance can occur no more than $\left\lfloor{{\textstyle{1\over 4}}n^2}\right\rfloor$ times in 4-D, where $\left\lfloor{x}\right\rfloor$ is the Floor Function.

References

Honsberger, R. Mathematical Gems II. Washington, DC: Math. Assoc. Amer., pp. 122-123, 1976.