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 3 of 8: Bound a chord from above
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In plain words

The circular geometry gives an upper chord bound.

XY=2R2−(R−12)2=2R−14<2nδXY=2\sqrt{R^2-(R-\frac12)^2}=2\sqrt{R-\frac14}<2\sqrt{n\delta}
Detailed analysis

Take the chord XYXY of the circle centered at BB with radius RR, perpendicular to ABAB, at distance 12\frac12 from AA. Pythagoras gives XY=2R2−(R−12)2=2R−14<2nδXY=2\sqrt{R^2-(R-\frac12)^2}=2\sqrt{R-\frac14}<2\sqrt{n\delta}.