Question 6
Prove that there exists a positive constant $c$ such that the following statement is true:
Consider an integer $n\gt1$, and a set $\mathcal S$ of $n$ points in the plane such that the distance between any two different points in $\mathcal S$ is at least 1. It follows that there is a line $\ell$ separating $\mathcal S$ such that the distance from any point of $\mathcal S$ to $\ell$ is at least $cn^{-1/3}$.
A line $\ell$ separates a set of points $\mathcal S$ if some segment joining two points in $\mathcal S$ crosses $\ell$.
Note: Weaker results with $cn^{-1/3}$ replaced by $cn^{-\alpha}$ may be awarded points depending on the value of the constant $\alpha\gt1/3$.
Consider an integer $n\gt1$, and a set $\mathcal S$ of $n$ points in the plane such that the distance between any two different points in $\mathcal S$ is at least 1. It follows that there is a line $\ell$ separating $\mathcal S$ such that the distance from any point of $\mathcal S$ to $\ell$ is at least $cn^{-1/3}$.
A line $\ell$ separates a set of points $\mathcal S$ if some segment joining two points in $\mathcal S$ crosses $\ell$.
Note: Weaker results with $cn^{-1/3}$ replaced by $cn^{-\alpha}$ may be awarded points depending on the value of the constant $\alpha\gt1/3$.