A necessary and sufficient condition for $k$-transversals
Daniel McGinnis, Nikola Sadovek
TL;DR
Problem: characterize when a finite family $\mathcal{F}$ of convex sets in $\mathbb{R}^d$ admits a $k$-transversal for all $0 \le k \le d-1$. The authors introduce the linear-algebraic notion of $\mathbb{F}$-dependency consistency with $(d-k)$-tuples in a finite set $P \subseteq \mathbb{F}^k$ and prove a necessary-and-sufficient condition that exactly characterizes the existence of a $k$-transversal, including a complex analogue for $\mathbb{F}=\mathbb{C}$. The proof uses a configuration-space/test-map scheme and a Borsuk-Ulam-type theorem on Stiefel manifolds, unifying Helly's theorem ($k=0$) and the Goodman–Pollack–Wenger theorem ($k=d-1$) and yielding complex-central transversal results. Consequently, central transversal theorems of Živaljević–Vrećica and Dol'nikov (real) and Sadovek–Soberón (complex) follow as corollaries. The work provides a topological framework for geometric transversal theory and suggests avenues for colorful generalizations and further extensions.
Abstract
We solve a long-standing open problem posed by Goodman \& Pollack in 1988 by establishing a necessary and sufficient condition for a family of convex sets in $\mathbb{R}^d$ to admit a $k$-transversal for any $0 \le k \le d-1$. This result is a common generalization of Helly's theorem ($k=0$) and the Goodman-Pollack-Wenger theorem ($k=d-1$). Additionally, we obtain an analogue in the complex setting by characterizing the existence of a complex $k$-transversal to a family of convex sets in $\mathbb{C}^d$, extending the work of McGinnis ($k=d-1$). Our approach is topological and employs a Borsuk-Ulam-type theorem on Stiefel manifolds. Finally, we demonstrate how our results imply the central transversal theorems of Živaljević-Vrećica and Dol'nikov in the real case and of Sadovek-Soberón in the complex case.