On the Geometry of Holmsen's Combinatorial Version of the Colorful Carath\'eodory

Carath'eodorys Theorem of convex hulls plays an important role in convex geometry. In 1982, B'ar'any formulated and proved a more general version, called the Colorful Carath'eodory. 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.