Geodesic
On the Equivalence of the Theorems Helly, Radon, and Carathéodory via Convex Analysis
Journal of convex analysis, Tome 22 (2015) no. 2, pp. 591-601
Cet article a éte moissonné depuis la source Heldermann Verlag

Voir la notice de l'article

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.
Classification : 52A05, 52A20, 52A35, 52A37
Mots-clés : Caratheodory's theorem, convex function, convex hull, distance function, Helly's theorem, Radon's theorem, subdifferential, subgradient 
@article{JCA_2015_22_2_JCA_2015_22_2_a14,
     author = {H. Martini and N. M. Nam and A. Robinson},
     title = {On the {Equivalence} of the {Theorems} {Helly,} {Radon,} and {Carath\'eodory} via {Convex} {Analysis}},
     journal = {Journal of convex analysis},
     pages = {591--601},
     year = {2015},
     volume = {22},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/JCA_2015_22_2_JCA_2015_22_2_a14/}
}
TY  - JOUR
AU  - H. Martini
AU  - N. M. Nam
AU  - A. Robinson
TI  - On the Equivalence of the Theorems Helly, Radon, and Carathéodory via Convex Analysis
JO  - Journal of convex analysis
PY  - 2015
SP  - 591
EP  - 601
VL  - 22
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/JCA_2015_22_2_JCA_2015_22_2_a14/
ID  - JCA_2015_22_2_JCA_2015_22_2_a14
ER  - 
%0 Journal Article
%A H. Martini
%A N. M. Nam
%A A. Robinson
%T On the Equivalence of the Theorems Helly, Radon, and Carathéodory via Convex Analysis
%J Journal of convex analysis
%D 2015
%P 591-601
%V 22
%N 2
%U http://geodesic.mathdoc.fr/item/JCA_2015_22_2_JCA_2015_22_2_a14/
%F JCA_2015_22_2_JCA_2015_22_2_a14
H. Martini; N. M. Nam; A. Robinson. On the Equivalence of the Theorems Helly, Radon, and Carathéodory via Convex Analysis. Journal of convex analysis, Tome 22 (2015) no. 2, pp. 591-601. http://geodesic.mathdoc.fr/item/JCA_2015_22_2_JCA_2015_22_2_a14/