  In this paper, we study the $d$-dimensional rectilinear drawings of the
complete $d$-uniform hypergraph $K_{2d}^d$. Anshu et al. [Computational
Geometry: Theory and Applications, 2017] used Gale transform and Ham-Sandwich
theorem to prove that there exist $\Omega \left(2^d\right)$ crossing pairs of
hyperedges in such a drawing of $K_{2d}^d$. We improve this lower bound by
showing that there exist $\Omega \left(2^d \sqrt{ d}\right)$ crossing pairs of
hyperedges in a $d$-dimensional rectilinear drawing of $K_{2d}^d$. We also
prove the following results.
  1. There are $\Omega \left(2^d {d^{3/2}}\right)$ crossing pairs of hyperedges
in a $d$-dimensional rectilinear drawing of $K_{2d}^d$ when its $2d$ vertices
are either not in convex position in $\mathbb{R}^d$ or form the vertices of a
$d$-dimensional convex polytope that is $t$-neighborly but not
$(t+1)$-neighborly for some constant $t\geq1$ independent of $d$.
  2. There are $\Omega \left(2^d {d^{5/2}}\right)$ crossing pairs of hyperedges
in a $d$-dimensional rectilinear drawing of $K_{2d}^d$ when its $2d$ vertices
form the vertices of a $d$-dimensional convex polytope that is
$(\lfloor{d/2}\rfloor-t')$-neighborly for some constant $t' \geq 0$ independent
of $d$.
