A necessary and sufficient condition for $k$-transversals

TL;DR

This paper provides a comprehensive characterization of when a family of convex sets in real and complex spaces admits a $k$-transversal, unifying several classical theorems through a topological approach.

Contribution

It establishes a necessary and sufficient condition for the existence of $k$-transversals, generalizing Helly's theorem and related results, using topological methods.

Findings

Unified condition for $k$-transversals in $ eal^d$

Extension of results to complex convex sets in $ield{C}^d$

Implications for central transversal theorems

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 \v{Z}ivaljevi\'c-Vre\'cica and Dol'nikov in the real case and of Sadovek-Sober\'on in the complex case.