Large Triangles in the d-Dimensional Unit-Cube

Abstract

Abstract. We consider a variant of Heilbronn's triangle problem by asking for fixed dimension d ≥ 2 for a distribution of n points in the d-dimensional unit-cube [0,1] d such that the minimum (2-dimensional) area of a triangle among these n points is maximal. Denoting this maximum value by Δ of f −line d (n) and Δ on−line d(n) for the off-line and the online situation, respectively, we will show that c1·(log n)

Distributions of Points and Large Convex Hulls of k Points

“…With the results from [14], i.e. , using a result of Ajtai, Komlós, Pintz, Spencer and Szemerédi [1] on uncrowded hypergraphs we can improve Theorem 1 slightly (Details are omitted here), namely for fixed integers d, k ≥ 3 and a fixed integer j 0 with 3 ≤ j 0 ≤ d + 1 one can find in polynomial time a configuration of n points in [0,1] d , such that, simultaneously for j = 2, .…”

“…Notice that for fixed integers d, j ≥ 2, Theorem 1 yields [14]. Somewhat surprisingly, achieving by a deterministic polynomial time algorithm for the same n points in [0,1] d the lower bound ∆ j,d (n) = Ω(1/n (j−1)/(1+|d−j+1|) ), simultaneously for j = 2, .…”

“…An extension of the range of k might also improve his lower bounds on the query complexity, see [7]. Here we consider the case of fixed values k and d. Areas of triangles arising from n points in [0, 1] d have been investigated in [12], where for fixed dimension d ≥ 2 it has been shown that…”

Distributions of Points in d Dimensions and Large k-Point Simplices

“…For k = d, the upper bound in (2) yields the bound from [8] and the lower bound in (1) gives the result from [13]. For k = 2 and any fixed dimension d ≥ 2, the bounds in (1) yield the above mentioned result from [14]. Indeed, our arguments for proving Theorem 1 give a randomized polynomial in n time algorithm, which finds a distribution of n points in [0, 1] d that achieves the lower bound in (1).…”

Distributions of Points in d Dimensions and Large k-Point Simplices

“…Here we consider the following generalization of Heilbronn's problem: given fixed integers d, k with 1 ≤ k ≤ d, find for every integer n ≥ k a distribution of n points in the d-dimensional unit cube [0,1] d such that the minimum volume of a (k + 1)-point simplex arising from these n points is as large as possible. Let 1] d , were given by this author in [14], where it has been shown that c 2,d · (log n)…”