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A complex analogue of the Goodman-Pollack-Wenger theorem

Daniel McGinnis

TL;DR

The paper establishes a complex analogue of the Goodman–Pollack–Wenger theorem by proving that a finite family of convex sets in $\mathbb{C}^d$ admits a complex $(d-1)$-transversal if and only if the family is dependency-consistent with a set $P\subset \mathbb{C}^{d-1}$. The key method adapts the GPW framework to the complex setting, formulating dependency-consistency as the complex counterpart to consistent separation and employing a Borsuk–Ulam argument on the sphere $S^{2d+1}$ with an odd map to derive the transversal from a zero. The result fills a gap in transversal theory for intermediate dimensions ($0<k<d-1$) by resolving the complex $(d-1)$-transversal case and providing a structurally analogous criterion to the real GPW theorem. This contributes a new bridge between complex convex geometry and topological methods, with potential implications for complex hyperplane transversal problems and related combinatorial-topological questions.

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.

A complex analogue of the Goodman-Pollack-Wenger theorem

TL;DR

The paper establishes a complex analogue of the Goodman–Pollack–Wenger theorem by proving that a finite family of convex sets in admits a complex -transversal if and only if the family is dependency-consistent with a set . The key method adapts the GPW framework to the complex setting, formulating dependency-consistency as the complex counterpart to consistent separation and employing a Borsuk–Ulam argument on the sphere with an odd map to derive the transversal from a zero. The result fills a gap in transversal theory for intermediate dimensions () by resolving the complex -transversal case and providing a structurally analogous criterion to the real GPW theorem. This contributes a new bridge between complex convex geometry and topological methods, with potential implications for complex hyperplane transversal problems and related combinatorial-topological questions.

Abstract

A \textit{-transversal} to family of sets in is a -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 -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 -transversals to finite families of convex sets in with no disjointness condition on the family of sets. We prove an analogue of the Goodman-Pollack-Wenger theorem in the complex setting.
Paper Structure (4 sections, 4 theorems, 13 equations, 1 figure)

This paper contains 4 sections, 4 theorems, 13 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Hyperplane transversals revisited
  3. Main result
  4. Acknowledgements

Key Result

Theorem 1.1

A finite family of pairwise disjoint convex sets in $\mathbb{R}^2$ has a $1$-transversal if and only if the sets in the family can be linearly ordered such that any three sets have a $1$-transversal consistent with the ordering.

Figures (1)

  • Figure 1: Depiction of $F_{\mathbf{x}_0}$, $p_{\mathbf{x}_0,F}\mathbf{x}_0$, and $\textrm{proj}_{\mathbf{x}_0}(p_F)$ in the complex plane $P_{\mathbf{x}_0}$.

Theorems & Definitions (7)

  • Theorem 1.1: Hadwiger HadwigerLines
  • Theorem 2.1: Goodman-Pollack-Wenger theorem PollackNecessary1990
  • Definition 3.1
  • Remark 3.2
  • Theorem 3.3: Borsuk-Ulam theorem
  • Theorem 3.4: Main theorem
  • proof