On the Geometry of Holmsen's Combinatorial Version of the Colorful Carathéodory

Abstract

Carathéodorys Theorem of convex hulls plays an important role in convex geometry. In 1982, Bárány formulated and proved a more general version, called the Colorful Carathéodory. This colorful version was even more generalized by Holmsen in 2016. He formulated a combinatorial extension and found a topological proof. Taking a dual point of view we gain an equivalent formulation of Holmsen's result that has a more geometric meaning.