Subfield: Number Theory


Question: Prove that there exists a positive constant $c$ such that the following statement is true:

Assume that $n$ is an integer with $n \geqslant 2$, and let $\mathcal{S}$ be a set of $n$ points in the plane such that the distance between any two distinct points in $\mathcal{S}$ is at least 1 . Then there is a line $\ell$ separating $\mathcal{S}$ such that the distance from any point of $\mathcal{S}$ to $\ell$ is at least $c n^{-1 / 3}$.

(A line $\ell$ separates a point set $\mathcal{S}$ if some segment joining two points in $\mathcal{S}$ crosses $\ell$.)

Solution:
We prove that the desired statement is true with $c=\frac{1}{8}$. Set $\delta=\frac{1}{8} n^{-1 / 3}$. For any line $\ell$ and any point $X$, let $X_{\ell}$ denote the projection of $X$ to $\ell$; a similar notation applies to sets of points.

Suppose that, for some line $\ell$, the set $\mathcal{S}_{\ell}$ contains two adjacent points $X$ and $Y$ with $X Y=2 d$. Then the line perpendicular to $\ell$ and passing through the midpoint of segment $X Y$ separates $\mathcal{S}$, and all points in $\mathcal{S}$ are at least $d$ apart from $\ell$. Thus, if $d \geqslant \delta$, then a desired line has been found. For the sake of contradiction, we assume that no such points exist, in any projection.

Choose two points $A$ and $B$ in $\mathcal{S}$ with the maximal distance $M=A B$ (i.e., $A B$ is a diameter of $\mathcal{S})$; by the problem condition, $M \geqslant 1$. Denote by $\ell$ the line $A B$. The set $\mathcal{S}$ is contained in the intersection of two disks $D_{A}$ and $D_{B}$ of radius $M$ centered at $A$ and $B$, respectively. Hence, the projection $\mathcal{S}_{\ell}$ is contained in the segment $A B$. Moreover, the points in $\mathcal{S}_{\ell}$ divide that segment into at most $n-1$ parts, each of length less than $2 \delta$. Therefore,

$$
M<n \cdot 2 \delta .
\tag{1}
$$

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Choose a point $H$ on segment $A B$ with $A H=\frac{1}{2}$. Let $P$ be a strip between the lines $a$ and $h$ perpendicular to $A B$ and passing through $A$ and $H$, respectively; we assume that $P$ contains its boundary, which consists of lines $a$ and $h$. Set $\mathcal{T}=P \cap \mathcal{S}$ and let $t=|\mathcal{T}|$. By our assumption, segment $A H$ contains at least $\left\lceil\frac{1}{2}:(2 \delta)\right\rceil$ points of $S_{\ell}$, which yields

$$
t \geqslant \frac{1}{4 \delta}\tag{2}
$$

Notice that $\mathcal{T}$ is contained in $Q=P \cap D_{B}$. The set $Q$ is a circular segment, and its projection $Q_{a}$ is a line segment of length

$$
2 \sqrt{M^{2}-\left(M-\frac{1}{2}\right)^{2}}<2 \sqrt{M}
$$

On the other hand, for any two points $X, Y \in \mathcal{T}$, we have $X Y \geqslant 1$ and $X_{\ell} Y_{\ell} \leqslant \frac{1}{2}$, so $X_{a} Y_{a}=$ $\sqrt{X Y^{2}-X_{\ell} Y_{\ell}^{2}} \geqslant \frac{\sqrt{3}}{2}$. To summarize, $t$ points constituting $\mathcal{T}_{a}$ lie on the segment of length less than $2 \sqrt{M}$, and are at least $\frac{\sqrt{3}}{2}$ apart from each other. This yields $2 \sqrt{M}>(t-1) \frac{\sqrt{3}}{2}$, or

$$
t<1+\frac{4 \sqrt{M}}{\sqrt{3}}<4 \sqrt{M}\tag{3}
$$



as $M \geqslant 1$.

Combining the estimates (1), (2), and (3), we finally obtain

$$
\frac{1}{4 \delta} \leqslant t<4 \sqrt{M}<4 \sqrt{2 n \delta}, \quad \text { or } \quad 512 n \delta^{3}>1
$$

which does not hold for the chosen value of $\delta$.