Concyclic intervals in the plane (Q6624190)

In this interesting paper the authors study the geometry of concyclic intervals in the plane. An interval in \(\mathbb{R}^{2}\) is a line segment which will be denote by a four-tuple \((a,b;c,d) \in \mathbb{R}^{4}\), where \((a,b), (c,d) \in \mathbb{R}^{2}\) denotes the coordinates of the endpoints. One requires that \((a,b) \neq (c,d)\). We say that a pair of distinct intervals \((a,b;c,d), (a',b';c'd') \in \mathbb{R}^{4}\) is concyclic if there is a single circle containing the four endpoints \((a,b),(a',b'), (c,d), (c'd')\). Note that the case where the radius of the inscribed circle is infinite is forbidden. The main question that one can formulate is the following.\N\NQuestion: Given a set of intervals for which many pairs are concyclic, what structure must the interval possess?\N\NWe say that a polynomial \(F \in \mathbb{R}[x,y]\) is a bicircular quartic if the leading homogeneous term is divisible by \((x^{2}+y^{2})^{2}\) and the degree \(3\) homogeneous term is divisible by \(x^{2}+y^{2}\). A circular cubic is a curve with degree \(3\) homogeneous term divisible by \(x^2 + y^2\). For the formulation of the main result of the paper under review, we need the notation \(A\gtrsim B\) that means that there exists a universal constant \(C>0\) for which \(A \geq CB\).\N\NMain Theorem. Let \(\mathfrak{I}\) be a set of \(N\) distinct intervals in \(\mathbb{R}^{2}\) with no pair intervals contained in a common line. If more than \(N^{3/2}\cdot\log N\) pairs of intervals are concyclic, then one of the following holds.\N\begin{itemize}\N\item[1)] There is a circle \(\mathcal{C}\) in \(\mathbb{R}^{2}\) such that \(\gtrsim N^{1/2}\) intervals have both endpoints on \(\mathcal{C}\).\N\item[2a)] \(\gtrsim N^{1/2}\) intervals are parallel and are perpendicularly bisected by a common line, or\N\item[2b)] after a translation of \(\mathfrak{I}\), there exists a constant \(h\geq 0\) such that \(\gtrsim N^{1/2}\) intervals \((a,b;c,d)\) satisfy \((a^{2}+b^{2})(c^{2}+d^{2}) = h\) and the lines containing these intervals are all concurrent.\N\item[3)] There are two subsets \(\mathfrak{I}_{1}, \mathfrak{I}_{2} \subset \mathfrak{I}\) such that for any \(I_{1} \in \mathfrak{I}_{1}\) and any \(I_{2} \in \mathfrak{I}_{2}\), the intervals \(I_{1}, I_{2}\) are concyclic. In addition, \(|\mathfrak{I}_{1}|\cdot |\mathfrak{I}_{2}| \gtrsim N\). Moreover, all endpoints of intervals in \(\mathfrak{I}_{1} \cup \mathfrak{I}_{2}\) lie on a bicircular quartic, circular cubic, or conic.\N\end{itemize}