An Almost Optimal Bound on the Number of Intersections of Two Simple Polygons

What is the maximum number of intersections of the boundaries of a simple m-gon and a simple n-gon? This is a basic question in combinatorial geometry, and the answer is easy if at least one of m and n is even: If both m and n are even, then every pair of sides may cross and so the answer is mn. If exactly one polygon, say the n-gon, has an odd number of sides, it can intersect each side of the m-gon polygon at most n−1 times; hence there are at most mn−m intersections. It is not hard to construct examples that meet these bounds. If both m and n are odd, the best known construction has mn−(m+n)+3 intersections, and it is conjectured that this is the maximum. However, the best known upper bound is only mn−(m+⌈n/6⌉), for m≥n. We prove a new upper bound of mn−(m+n)+C for some constant C, which is optimal apart from the value of C

An Almost Optimal Bound on the Number of Intersections of Two Simple Polygons

What is the maximum number of intersections of the boundaries of a simple m-gon and a simple n-gon, assuming general position? This is a basic question in combinatorial geometry, and the answer is easy if at least one of m and n is even. If both m and n are odd, the best known construction has mn-(m+n)+3 intersections, and it is conjectured that this is the maximum. However, the best known upper bound is only mn-(m + ? n/6 ?), for m ? n. We prove a new upper bound of mn-(m+n)+C for some constant C, which is optimal apart from the value of C

An almost optimal bound on the number of intersections of two simple polygons

What is the maximum number of intersections of the boundaries of a simple $m$-gon and a simple $n$-gon, assuming general position? This is a basic question in combinatorial geometry, and the answer is easy if at least one of $m$ and $n$ is even: If both $m$ and $n$ are even, then every pair of sides may cross and so the answer is $mn$. If exactly one polygon, say the $n$-gon, has an odd number of sides, it can intersect each side of the $m$-gon at most $n-1$ times; hence there are at most $mn-m$ intersections. It is not hard to construct examples that meet these bounds. If both $m$ and $n$ are odd, the best known construction has $mn-(m+n)+3$ intersections, and it is conjectured that this is the maximum. However, the best known upper bound is only $mn-(m + \lceil \frac{n}{6} \rceil)$, for $m \ge n$. We prove a new upper bound of $mn-(m+n)+C$ for some constant $C$, which is optimal apart from the value of $C$.Comment: 18 pages, 24 figures. To appear in the proceedings of the 36th International Symposium on Computational Geometry (SoCG 2020) in June 2016 (Eds. Sergio Cabello and Danny Chen

An Almost Optimal Bound on the Number of Intersections of Two Simple Polygons

What is the maximum number of intersections of the boundaries of a simple m-gon and a simple n-gon? This is a basic question in combinatorial geometry, and the answer is easy if at least one of m and n is even: If both m and n are even, then every pair of sides may cross and so the answer is mn. If exactly one polygon, say the n-gon, has an odd number of sides, it can intersect each side of the m-gon polygon at most n−1 times; hence there are at most mn−m intersections. It is not hard to construct examples that meet these bounds. If both m and n are odd, the best known construction has mn−(m+n)+3 intersections, and it is conjectured that this is the maximum. However, the best known upper bound is only mn−(m+⌈n/6⌉), for m≥n. We prove a new upper bound of mn−(m+n)+C for some constant C, which is optimal apart from the value of C

An almost optimal bound on the number of intersections of two simple polygons

What is the maximum number of intersections of the boundaries of a simple m-gon and a simple n-gon, assuming general position? This is a basic question in combinatorial geometry, and the answer is easy if at least one of m and n is even. If both m and n are odd, the best known construction has mn − (m + n) + 3 intersections, and it is conjectured that this is the maximum. However, the best known upper bound is only mn − (m + dn6 e), for m ≥ n. We prove a new upper bound of mn − (m + n) + C for some constant C, which is optimal apart from the value of C. © Eyal Ackerman, Balázs Keszegh, and Günter Rote; licensed under Creative Commons License CC-BY 36th International Symposium on Computational Geometry (SoCG 2020)