# Perron and Frobenius meet Carath\'{e}odory

## TL;DR

This paper introduces a novel method leveraging the Perron-Frobenius Theorem to prove Carathéodory-type theorems, potentially opening new avenues for related geometric results.

## Contribution

It presents a new approach using matrix theory to prove classical geometric theorems, offering a fresh perspective and potential extensions.

## Key findings

- New proof technique for Carathéodory-type theorems
- Potential extension to Colourful Carathéodory Theorem
- Open question on broader applicability

## Abstract

We present a new approach of proving certain Carath\'{e}odory-type theorems using the Perron-Frobenius Theorem, a classical result in matrix theory describing the largest eigenvalue of a matrix with positive entries. One of the problems left open in this note is whether our approach may be extended to prove similar results in the area, in particular the Colourful Carath\'{e}odory Theorem.