A necessary and sufficient condition for $k$-transversals

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.