On colorful generalizations of the Goodman--Pollack transversal problem

TL;DR

This paper introduces topological methods to establish colorful and matroidal solutions for the Goodman-Pollack transversal problem, unifying and extending several classical theorems in convex geometry.

Contribution

It provides a novel topological framework using matroidal joins and homotopy colimits to solve and generalize transversal theorems in convex geometry.

Findings

Unified several classical and recent theorems in convex geometry.

Extended the colorful Helly theorem and Goodman-Pollack-Wenger theorem to matroidal settings.

Developed new topological tools like matroidal joins and derived connectivity estimates.

Abstract

We establish a colorful and, more generally, matroidal solution to the problem of Goodman and Pollack on the existence of an $\mathbb{F}$-affine $k$-dimensional transversal to a family of convex sets in $\mathbb{F}^d$, where $0 \le k \le d - 1$ is an integer and $\mathbb{F} \in \{\mathbb{R}, \mathbb{C}\}$ is a field. Our results unify several classical and recent theorems. In the case $k=0$, we recover the colorful Helly theorem of Lov\'asz, together with a matroidal extension due to Kalai and Meshulam. In the opposite extremal case $k=d-1$, we obtain Holmsen's colorful and matroidal generalization of the Goodman-Pollack-Wenger theorem. Additionally, we extend the recent noncolorful solution of the Goodman-Pollack problem by McGinnis and the author. As the main application, we obtain a matroidal and colorful Dol'nikov-type transversal theorem. Our methods are topological. We introduce matroidal joins, defined as homotopy colimits of diagrams over face posets of matroidal complexes, and derive estimates on their connectivity. The proof additionally relies on adaptations of nonexistence results for equivariant maps from Stiefel manifolds to spheres.