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 6 of 8: Bound the same chord from below
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In plain words

Separated projections consume chord length.

XY>32(#(T∩S)−1)>34δ−32XY>\frac{\sqrt3}{2}(\#(T\cap S)-1)>\frac{\sqrt3}{4\delta}-\frac{\sqrt3}{2}
Detailed analysis

Ordering the projections of the points of S∩TS\cap T on XYXY gives XY>32(#(T∩S)−1)>34δ−32XY>\frac{\sqrt3}{2}(\#(T\cap S)-1)>\frac{\sqrt3}{4\delta}-\frac{\sqrt3}{2}.