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

## TL;DR

This paper explores the geometric aspects of Holmsen's combinatorial extension of the Colorful Carathéodory theorem, providing a dual perspective that enhances understanding of its geometric implications.

## Contribution

It offers a dual geometric formulation of Holmsen's combinatorial version, deepening the understanding of its geometric structure.

## Key findings

- Dual formulation reveals new geometric insights
- Enhances understanding of Holmsen's combinatorial extension
- Provides a more geometric perspective on the theorem

## Abstract

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.