Eppstein's bound on intersecting triangles revisited
نویسندگان
چکیده
Let S be a set of n points in the plane, and let T be a set of m triangles with vertices in S. Then there exists a point in the plane contained in Ω(m/(n log n)) triangles of T . Eppstein (1993) gave a proof of this claim, but there is a problem with his proof. Here we provide a correct proof by slightly modifying Eppstein’s argument.
منابع مشابه
Arrangements of Pseudocircles: Triangles and Drawings
A pseudocircle is a simple closed curve on the sphere or in the plane. The study of arrangements of pseudocircles was initiated by Grünbaum, who defined them as collections of simple closed curves that pairwise intersect in exactly two crossings. Grünbaum conjectured that the number of triangular cells p3 in digon-free arrangements of n pairwise intersecting pseudocircles is at least 2n−4. We p...
متن کاملSimilar Triangles, Another Trace of the Golden Ratio
In this paper we investigate similar triangles which are not congruent but have two pair congruent sides. We show that greatest lower bound of values for similarity ratio of such triangles is golden ratio. For right triangles, we prove that the supremum of values for similarity ratio is the square root of the golden ratio.
متن کاملImproved Bounds for Intersecting Triangles and Halving Planes
If a configuration of m triangles in the plane has only n points as vertices, then there must be a set of max { dm/(2n− 5)e Ω(m3/(n6 log n)) triangles having a common intersection. As a consequence the number of halving planes for a three-dimensional point set is O(n log n). For all m and n there exist configurations of triangles in which the largest common intersection involves max {dm/(2n− 5)...
متن کاملArrangements of Pseudocircles: On Circularizability
An arrangement of pseudocircles is a collection of simple closed curves on the sphere or in the plane such that every pair is either disjoint or intersects in exactly two crossing points. We call an arrangement intersecting if every pair of pseudocircles intersects twice. An arrangement is circularizable if there is a combinatorially equivalent arrangement of circles. Kang and Müller showed tha...
متن کاملImproving Lower Bound on Opaque Set for Equilateral Triangle
An opaque set (or a barrier) for U ⊆ R is a set B of finite-length curves such that any line intersecting U also intersects B. In this paper, we consider the lower bound for the shortest barrier when U is the unit equilateral triangle. The known best lower bound for triangles is the classic one by Jones [9], which exhibits that the length of the shortest barrier for any convex polygon is at lea...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- J. Comb. Theory, Ser. A
دوره 116 شماره
صفحات -
تاریخ انتشار 2009