Abstract
In this paper, we study the d-dimensional rectilinear drawings of the complete d-uniform hypergraph K2dd. Anshu et al. (2017) [3] used Gale transform and Ham-Sandwich theorem to prove that there exist Ω(2d) crossing pairs of hyperedges in such a drawing of K2dd. We improve this lower bound by showing that there exist Ω(2dd) crossing pairs of hyperedges in a d-dimensional rectilinear drawing of K2dd. We also prove the following results.1.There are Ω(2dd3/2) crossing pairs of hyperedges in a d-dimensional rectilinear drawing of K2dd when its 2d vertices are either not in convex position in Rd or form the vertices of a d-dimensional convex polytope that is t-neighborly but not (t+1)-neighborly for some constant t≥1 independent of d.2.There are Ω(2dd5/2) crossing pairs of hyperedges in a d-dimensional rectilinear drawing of K2dd when its 2d vertices form the vertices of a d-dimensional convex polytope that is (⌊d/2⌋−t′)-neighborly for some constant t′≥0 independent of d.
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