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 5 of 8: Project points in T onto the chord
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In plain words

The thin width forces strong separation along the chord.

∣P−Q∣≥1,∣proj⁡XY(P)−proj⁡XY(Q)∣>32(P,Q∈S∩T)|P-Q|\ge1,\quad |\operatorname{proj}_{XY}(P)-\operatorname{proj}_{XY}(Q)|>\frac{\sqrt3}{2}\quad(P,Q\in S\cap T)
Detailed analysis

For two points P,Q∈S∩TP,Q\in S\cap T, their coordinate difference in the ABAB direction is less than 12\frac12. If their projections onto XYXY were at most 32\frac{\sqrt3}{2} apart, Pythagoras would make PQ<1PQ<1, contradicting the separation of SS. Hence successive projections onto XYXY differ by more than 32\frac{\sqrt3}{2}.