On the Equivalence of the Theorems Helly, Radon, and Carathéodory via Convex Analysis

Helly's theorem is an important result from Convexity and Combinatorial 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 has close connections to two other well-known theorems: Radon's theorem and Carathéeodory's theorem. In this paper 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 new proof of Helly's theorem, and in addition we show in a complete way that these three theorems are equivalent in the sense that using one of them allows us to derive the others.