Date of Award
Mau Nam Nguyen
Convex geometry, Convex domains, Mathematical analysis, Mathematical optimization, Discrete geometry
Helly's theorem is an important result from Convex Geometry. It gives sufficient conditions for a family of convex sets to have a nonempty intersection. A large variety of proofs as well as applications are known. Helly's theorem also has close connections to two other well-known theorems from Convex Geometry: Radon's theorem and Carathéodory's theorem. In this project we study Helly's theorem and its relations to Radon's theorem and Carathéodory's theorem by using tools of Convex Analysis and Optimization. More precisely, we will give a novel proof of Helly's theorem, and in addition we show in a complete way that these three famous theorems are equivalent in the sense that using one of them allows us to derive the others.
Robinson, Adam L., "Helly's Theorem and Its Equivalences via Convex Analysis" (2014). University Honors Theses. Paper 67.