
Journal of Convex Analysis 22 (2015), No. 2, 591601 Copyright Heldermann Verlag 2015 On the Equivalence of the Theorems Helly, Radon, and Carathéodory via Convex Analysis Horst Martini Fakultät für Mathematik, Technische Universität, 09107 Chemnitz, Germany horst.martini@mathematik.tuchemnitz.de Nguyen Mau Nam Fariborz Maseeh Dept. of Mathematics and Statistics, Portland State University, PO Box 751, Portland, OR 97207, U.S.A. mau.nam.nguyen@pdx.edu Adam Robinson Fariborz Maseeh Dept. of Mathematics and Statistics, Portland State University, PO Box 751, Portland, OR 97207, U.S.A. ad7@pdx.edu 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 wellknown 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. Keywords: Caratheodory's theorem, convex function, convex hull, distance function, Helly's theorem, Radon's theorem, subdifferential, subgradient. MSC: 52A05, 52A20, 52A35, 52A37 [ Fulltextpdf (134 KB)] for subscribers only. 