The average‐case area of Heilbronn‐type triangles*

Abstract

From among n 3 triangles with vertices chosen from n points in the unit square, let T be the one with the smallest area, and let A be the area of T . Heilbronn's triangle problem asks for the maximum value assumed by A over all choices of n points. We consider the average-case: If the n points are chosen independently and at random (with a uniform distribution), then there exist positive constants c and C such that c/n 3

Distributions of Points and Large Quadrangles

“…Upper bounds on ∆ 3 (n) were given by Roth [14][15][16][17][18] and Schmidt [19] and, improving these earlier results, the currently best upper bound ∆ 3 (n) = O(2 c √ log n /n 8/7 ), where c > 0 is a constant, is due to Komlós, Pintz and Szemerédi [10]. Recently, Jiang, Li and Vitany [9] showed by using methods from Kolmogorov complexity that if n points are distributed uniformly at random and independently of each other in the unit-square [0, 1] 2 , then the expected value of the minimum area of a triangle formed by some three of these n random points is equal to Θ(1/n 3 ). As indicated in [9], this result might be of use to measure the affiancy of certain Monte Carlo methods for determining fair market values of derivatives.…”

“…With (1), (2), (4), (5), (9) and using that A > 1, the running time of this derandomization is given by…”

“…Recently, Jiang, Li and Vitany [9] showed by using methods from Kolmogorov complexity that if n points are distributed uniformly at random and independently of each other in the unit-square [0, 1] 2 , then the expected value of the minimum area of a triangle formed by some three of these n random points is equal to Θ(1/n 3 ). As indicated in [9], this result might be of use to measure the affiancy of certain Monte Carlo methods for determining fair market values of derivatives. Higher dimensional extensions of Heilbronn's triangle problem were investigated by Barequet [2,3], who considered for fixed integers d ≥ 2 the minimum volumes of simplices among n points in the d-dimensional unit-cube [0, 1] d , maximized over all distributions of n points in [0,1] d , see also [12], [13] and Brass [6].…”

Point sets in the unit square and large areas of convex hulls of subsets of points

“…has been shown in [12] to be equal to ¢(1an 3 ). Variants of Heilbronn's triangle problem in higher dimensions were investigated by Barequet [3,4], who considered the minimum volumes of simplices among n points in the d-dimensional unit cube [0Y 1] d , see also [15] and Brass [8].…”

Large Triangles in the d-Dimensional Unit-Cube

“…From the other side, improving earlier results of Roth [14][15][16][17][18] and Schmidt [19], Komlós, Pintz and Szemerédi [10] proved the upper bound Δ 2 (n) = O(2 c √ log n /n 8/7 ) for some constant c > 0. Recently, for n randomly chosen points in the unit-square [0,1] 2 , the expected value of the minimum area of a triangle among these n points was determined to Θ(1/n 3 ) by Jiang, Li and Vitany [9]. A variant of Heilbronn's problem considered by Barequet [2] asks, given a fixed integer d ≥ 2, for the maximum value Δ * d (n) such that there exists a distribution of n points in the d-dimensional unit-cube [0,1] d where the minimum volume of a simplex formed by some (d + 1) of these n points is equal to Δ Here we will investigate the following extension of Heilbronn's problem to higher dimensions: for fixed integers d ≥ 2 and any given integer n ≥ 3 find a set of n points in the d-dimensional unit-cube [0,1] d such that the minimum area of a triangle determined by three of these n points is maximal.…”