MathLabs

Problem 6

Consider an integer n>1n>1, and a set SS of nn points in the plane such that the distance between any two different points in SS is at least 11. Prove there is a line ℓ\ell separating SS such that the distance from any point of SS to ℓ\ell is at least cn−1/3c n^{-1/3} for some absolute constant c>0c>0. (A line ℓ\ell separates a set of points SS if some segment joining two points in SS crosses ℓ\ell.)
Step 7 of 8: Combine the two bounds
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In plain words

The competing estimates force the cubic-root scale.

34δ−32<XY<2nδ⟹δ≥c′n−1/3(c′>0 absolute)\frac{\sqrt3}{4\delta}-\frac{\sqrt3}{2}<XY<2\sqrt{n\delta}\Longrightarrow \delta\ge c'n^{-1/3}\quad(c'>0\text{ absolute})
Detailed analysis

Combining 34δ−32<XY<2nδ\frac{\sqrt3}{4\delta}-\frac{\sqrt3}{2}<XY<2\sqrt{n\delta} gives δ≥c′n−1/3\delta\ge c'n^{-1/3} for an absolute constant c′>0c'>0: if δ≥1\delta\ge1 this is immediate, while for δ<1\delta<1 the left side is at least 38δ\frac{\sqrt3}{8\delta} and rearranging gives the cubic-root bound.