Xingyi Chen: En beviskedja i konvex geometri: från Radons lemma till en svag version av Center Transversal Theorem

Abstract: The main theme of this thesis is how local conditions on point sets and convex sets in R^d can give global geometric conclusions. We treat four results: Radon's lemma, Helly's theorem, the Centerpoint theorem and a weak version of the Center Transversal Theorem, and present them as a single coherent proof chain:
Radon's lemma ⇒ Helly's theorem ⇒ Centerpoint theorem ⇒ weak Center Transversal Theorem
The weak version of the Center Transversal Theorem is proved within the framework of the affine and convex geometry developed in the thesis, while the full version requires more advanced topological methods and therefore falls outside the scope of the thesis. The contribution of the thesis is to show a coherent chain of proofs and ideas that ties the four results together. Each of the first three results is used in the proof of the next, and the weak version of the Center Transversal Theorem forms the endpoint of the chain. The thesis is mainly written in affine and convex language and uses only elementary linear algebra, so the reader does not need a deep background in linear algebra to follow the arguments