Planar Distance

For $n$ points in the Plane, there are at least

$$N_1=\sqrt{n-{\textstyle{3\over 4}}}-{\textstyle{1\over 2}}$$

different Distances. The minimum Distance can occur only $\leq 3n-6$ times, and the Maximum Distance can occur $\leq n$ times. Furthermore, no Distance can occur as often as

$$N_2={\textstyle{1\over 4}}n(1+\sqrt{8n-7}\,)<{n^{3/2}\over\sqrt{2}}-{n\over 4}$$

times. No set of $n>6$ points in the Plane can determine only Isosceles Triangles.

See also Distance

References

Honsberger, R. ``The Set of Distances Determined by $n$ Points in the Plane.'' Ch. 12 in Mathematical Gems II. Washington, DC: Math. Assoc. Amer., pp. 111-135, 1976.