A complex analogue of the Goodman-Pollack-Wenger theorem

A $k\mathit{\text{-transversal}}$ to family of sets in ${\mathbb{R}}^d$ is a $k$-dimensional affine subspace that intersects each set of the family. In 1957 Hadwiger provided a necessary and sufficient condition for a family of pairwise disjoint, planar convex sets to have a $1$-transversal. After a series of three papers among the authors Goodman, Pollack, and Wenger from 1988 to 1990, Hadwiger’s Theorem was extended to necessary and sufficient conditions for $(d-1)$-transversals to finite families of convex sets in ${\mathbb{R}}^d$ with no disjointness condition on the family of sets. We prove an analogue of the Goodman-Pollack-Wenger theorem in the complex setting.