Helly, Radon, and Carathéodory type theorems. (English) Zbl 0791.52009

Eckhoff, Jürgen

Gruber, P. M. (ed.) et al., Handbook of convex geometry. Volume A. Amsterdam: North-Holland. 389-448 (1993).

This is a comprehensive survey article on generalizations and applications of theorems of Helly, Carathéodory and Radon. We obtain a very useful up-to-date source of information after 20 years that passed since the classical survey by L. Danzer, B. Grünbaum and V. Klee [Proc. Symp. Pure Math. 7, 101-180 (1963; Zbl 0132.174)]. The article falls into two parts. The first part deals with the combinatorial properties of families of convex sets. Here are the most important subjects: Helly type theorems, fixing and coloring properties, common transversals, intersection patterns of convex sets. The second part reviews convexity properties such as separation properties and generation of convex hulls. It also discusses theorems of Carathéodory, Radon, Tverberg, Steinitz, Kirchberger, Krasnosel’skij, and also some of their generalizations. The article deals mostly with results in the \(d\)- dimensional real linear space (equipped with Euclidean metric in a few cases). Numerous analogs of the Helly theorem obtained in graph theory and combinatorics are not considered.
For the entire collection see [Zbl 0777.52001].

Reviewer: M.Lassak (Bydgoszcz)

MSC:

52A35

Helly-type theorems and geometric transversal theory

Keywords:

Helly’s theorem; Carathéodory; Radon; convex sets; common transversals; intersection; separation; convex hulls

Citations:

Zbl 0777.52001; Zbl 0132.174