Problem 6

Consider an integer

n>1

, and a set

S

n

points in the plane such that the distance between any two different points in

S

is at least

1

. Prove there is a line

\ell

separating

S

such that the distance from any point of

S

\ell

is at least

c n^{-1/3}

for some absolute constant

c>0

. (A line

\ell

separates a set of points

S

if some segment joining two points in

S

crosses

\ell

Step 4 of 8: Count points in the thin segment

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In plain words

Many separated points must enter the narrow segment.

T=\text{the smaller segment cut off by }XY,\qquad \#(T\cap S)>\frac1{2\delta}

Detailed analysis

Let $T$ be the smaller circular segment cut off by $XY$ . Its width in the $AB$ direction is $\frac12$ . Since consecutive projections onto $AB$ are at most $\delta$ apart, the number of points of $S$ in this segment satisfies $\#(T\cap S)>\frac1{2\delta}$ .