In this paper, we consider a generalized version of the rectilinear crossing number problem of drawing complete graphs on a plane. The minimum number of crossing pairs of hyperedges in the d-dimensional rectilinear drawing of a d-uniform hypergraph is known as the d-dimensional rectilinear crossing number of the hypergraph. The currently best-known lower bound on the d-dimensional rectilinear crossing number of a complete d-uniform hypergraph with n vertices in general position in R d is Ω( 2 d √ d log d) n 2d . In this paper, we improve this lower bound to Ω(2 d ) n 2d . We also consider the special case when all the vertices of a d-uniform hypergraph are placed on the d-dimensional moment curve. For such complete d-uniform hypergraphs with n vertices, we show that the number of crossing pairs of hyperedges is Θ( 4 d √ d ) n 2d .

In this paper, we study the d-dimensional rectilinear drawings of the complete d-uniform hypergraph K d 2d . Anshu et al. [Computational Geometry: Theory and Applications, 2017] used Gale transform and Ham-Sandwich theorem to prove that there exist Ω 2 d crossing pairs of hyperedges in such a drawing of K d 2d . We improve this lower bound by showing that there exist Ω 2 d √ d crossing pairs of hyperedges in a ddimensional rectilinear drawing of K d 2d .In addition, we show that there are Ω 2 d d 3/2 crossing pairs of hyperedges in a d-dimensional rectilinear drawing of K d 2d when its 2d vertices are either not in convex position in R d or form the vertices of a ddimensional convex polytope satisfying one of the following two conditions: (a) the polytope is t-neighborly but not (t + 1)-neighborly for some constant t ≥ 1 independent of d, (b) the polytope is (⌊d/2⌋ − t ′ )neighborly for some constant t ′ ≥ 0 independent of d.

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