On the Number of Ordinary Circles Determined by n Points

Abstract

In this paper we deal with the following problem: Given a set B consisting of n points, not all on a line or a circle. A circle passes through exactly three points of B is called an ordinary circle. What is the minimal possible number of ordinary circles determined by B? In this paper we improve the best-known lower bound 22 247 n 2 to 1 9 n 2 .

On Sets Defining Few Ordinary Circles

“…The key tool in our proof is circle inversion , as it was in the earlier papers [ 1 , 12 , 26 ] on the ordinary circles problem; the first to use circle inversion in Sylvester–Gallai problems was Motzkin [ 18 ]. The simple reason for the relevance of circle inversion is that if we invert in a point of the given set, an ordinary circle through that point is turned into an ordinary line.…”

“…He suggested, cautiously, that the optimal bound is . Elliott’s result was improved by Bálintová and Bálint [ 1 , Rem., p. 288] to , and Zhang [ 26 ] obtained . Zhang also gave constructions of point sets on two concentric circles with ordinary circles.…”

“…We make no attempt to specify the threshold implicit in the phrase ‘for sufficiently large n ’; any improvement would depend on an improvement of the threshold in the result of Green and Tao [ 13 ]. For small n , the bound due to Zhang [ 26 ] remains the best known lower bound on the number of ordinary circles.…”

“…(Throughout the paper, by O(f (n)) we mean a function g(n) such that 0 g(n) Cf (n) for some constant C > 0 and all sufficiently large n. Thus, −O(n) is a function g(n) satisfying −Cn g(n) 0 for sufficiently large n.) He suggested, cautiously, that the optimal bound is 1 6 n 2 − O(n). Elliott's result was improved by Bálintová and Bálint [1, Remark, p. 288] to 11 247 n 2 − O(n), and Zhang [26] obtained 1 18 n 2 − O(n). Zhang also gave constructions of point sets on two concentric circles with 1 4 n 2 − O(n) ordinary circles.…”

Ordinary hyperspheres and spherical curves

“…The paper [7] also considered the alternative, more straightforward, definition of ordinary circles as only being proper circles excluding lines, as this was how the ordinary circles problem has been formulated originally [1,2,13]. However, this definition turned out to be less natural, as the class of proper circles are not invariant under inversions, and in the general approach of [7] the more inclusive definition had to be considered first.…”