Large triangles in the d-dimensional unit cube

Abstract

We consider a variant of Heilbronn's triangle problem by asking for a distribution of n points in the d-dimensional unit cube [ 0 , 1 ] d such that the minimum (two-dimensional) area of a triangle among these n points is as large as possible. Denoting by Δ d off - line ( n ) and Δ d on - line ( n ) the supremum of the minimum area of a triangle among n points over all distributions of n points in [ 0 , 1 ] d for the off-line and the on-line situation, respectively, for fixed dimension d ⩾ 2 we show that c 1 · ( log n ) 1 / ( d - 1 ) / n 2 / ( d - 1 ) ⩽ Δ d off - line ( n ) ⩽ c 1 ′ / n 2 / d and c 2 / n 2 / ( d - 1 ) ⩽ Δ d on - line ( n ) ⩽ c 2 ′ / n 2 / d for constants c 1 , c 2 , c 1 ′ , c 2 ′ > 0 which depend on d only. Moreover, we provide a deterministic polynomial time algorithm that achieves the lower bound Ω ( ( log n ) 1 / ( d - 1 ) / n 2 / ( d - 1 ) ) on the minimum area of a triangle among n points in [ 0 , 1 ] d in the off-line case.