In this paper we show lower bounds for the on-line version of Heilbronn's 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/n~(3.333...)) (resp., Ω(1/n~(5.292...))).
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