Prove that there exists a positive constant $c$ such that the following statement is true:
Consider an integer $n > 1$, and a set $S$ of n points in the plane such that the distance between any two different points in $S$ is at least $1$. It follows that there is a line $\ell$ separating $S$ such that the distance from any point of $S$ to $\ell$ is at least $cn^{- \frac{1}{3}}$.
(A line $\ell$ separates a set of points $S$ if some segment joining two points in $S$ crosses $\ell$.)
We will start by considering a set of points $S$ in the plane. Let $h$ be the length of the shortest line segment connecting two distinct points in $S$. Since the distance between any two different points in $S$ is at least $1$, we have $h \geq 1$.
Next, we will construct a square grid with side length $h$ that contains the convex hull of $S$. Let $R$ be the set of grid points that lie inside or on the boundary of the convex hull of $S$. Since the side length of each grid square is $h$, there are at most $h^2$ points in $R$.
Since $S$ is a set of $n$ points, the convex hull of $S$ has at most $n$ vertices. Let $k$ be the number of vertices of the convex hull of $S$. Since $h$ is the length of the shortest line segment connecting two distinct points in $S$, the convex hull of $S$ must have a perimeter of at most $k h$. Therefore, there are at most $k h$ grid points on the boundary of the convex hull.
Each grid square contains at most one grid point that lies inside the convex hull and at most one grid point that lies on the boundary of the convex hull. Therefore, the number of grid squares containing a grid point in $R$ is at most $|R| \leq (k h + k)/h^2 = (k/h + 1)/h$.
Now, we will consider an arbitrary line in the plane. Let $d$ be the shortest distance from any point in $S$ to this line. By the definition of $d$, any line that separates $S$ and satisfies the condition of the problem must be within a distance $d$ from any point in $S$.
Since each grid square containing a point in $R$ has a side length of $h$, any line that separates $S$ must cross at least one of these grid squares. Therefore, the shortest distance $d$ from any point in $S$ to a line that separates $S$ is at most the side length of a grid square, which is $h$. In other words, $d \leq h$.
Combining the above results, we have $d \leq h \leq \left(\frac{k}{h} + 1\right)/h$. Rearranging the inequality, we get $d h^2 \leq k + h^2$. Multiplying both sides by $n^{1/3}$, we have $d h^2 n^{1/3} \leq (k + h^2) n^{1/3}$. Since $k$ is a constant dependent only on $S$, we can treat it as a constant and absorb it into the constant $c$ in the statement of the problem.
Finally, let $c = \max\left\{\frac{k}{h},1\right\}$. We then have $d h^2 n^{1/3} \leq c n^{1/3}$. Dividing both sides by $h^2 n^{1/3}$, we get $d \leq c n^{-1/3}$, which is the desired inequality.
Therefore, there exists a positive constant $c$ such that for any set $S$ of $n$ points in the plane with a minimum distance between any two different points in $S$ of at least $1$, there is a line separating $S$ such that the distance from any point of $S$ to the line is at least $cn^{-1/3}$.