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 to admit a k -transversal for any . This result is a common generalization of Helly's theorem ( ) and the Goodman-Pollack-Wenger theorem ( ). 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 , extending the work of McGinnis ( ). 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.