A complex analogue of the Goodman-Pollack-Wenger theorem

TL;DR

This paper extends a classical geometric theorem about transversals of convex sets from real to complex spaces, providing a new analogue in the complex setting.

Contribution

It introduces a complex analogue of the Goodman-Pollack-Wenger theorem, broadening the understanding of transversals in complex Euclidean spaces.

Findings

Established a complex analogue of the Goodman-Pollack-Wenger theorem.

Extended classical real-space results to complex spaces.

Provided necessary and sufficient conditions for complex transversals.

Abstract

A \textit{$k$-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.