The on-line Heilbronnʹs triangle problem Original Research Article

In this paper, we show lower bounds for the on-line version of Heilbronnʹs triangle problem in three and four dimensions. Specifically, we provide incremental constructions for positioning n points in the 3-dimensional (resp., 4-dimensional) unit cube, for which every tetrahedron (resp., pentahedron) defined by four (resp., five) of these points has volume Ω(1/n3.333…) (resp., Ω(1/n5.292…)).