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Tverberg’s theorem is 50 years old: a survey. (English) Zbl 1401.52012

The theorem of H. Tverberg [J. Lond. Math. Soc. 41, 123–128 (1966; Zbl 0131.20002)] says that any set of \((r-1)(d+1)+1\) points in \({\mathbb R}^d\) can be partitioned into \(r\) parts whose convex hulls intersect. This result has had, and still has, considerable impact on combinatorial convexity. The present paper gives a survey, with emphasis on the developments of the last two decades. It starts with two modern proofs of Tverberg’s theorem, due respectively to J.-P. Roudneff [Eur. J. Comb. 22, No. 5, 745–765 (2001; Zbl 1007.52006)] and K. S. Sarkaria [Isr. J. Math. 79, No. 2–3, 317–320 (1992; Zbl 0786.52005)]. Section 2 deals with topological versions and relations to the generalized Van Kampen-Flores Theorem. The Topological Tverberg Theorem says: If \(f\) is a continuous map from the \(d\)-skeleton of the \(n\)-simplex \(\Delta^n\) to \({\mathbb R}^d\), with \(n=(r-1)(d+1)\) and \(r\) a prime power, then there are disjoint faces \(F_1,\dots,F_r\) of \(\Delta^n\) such that \(\bigcap_{j=1}^r f(F_j)\not=\emptyset\). After it was an open problem for decades, it was proved recently that this theorem does not hold if \(r\) is not a prime power. Section 3 treats various colorful versions and extensions of Tverberg’s theorem, and also mentions some open problems. Section 4 deals with several questions on the structure of Tverberg partitions, such as their minimal number, resistance to changes, modifications of the intersection condition, conjectures by Sierksma, Tverberg-Vrećica, Reay, and some new conjectures and open problems. Section 5 continues with variations and conjectures around Tverberg’s theorem and treats, in particular, universal Tverberg partitions. Various applications of Tverberg’s theorem to combinatorial geometry are the topic of Section 6, for example, the weak \(\varepsilon\)-net theorem for convex sets and the relation of Tverberg type results to Kneser hypergraphs. Section 7 is concerned with variations of Tverberg’s theorem if the underlying setting is changed; examples are integer coordinates, convexity spaces, quantitative versions. Altogether, this is a comprehensive and attractive survey on a great theorem and its aftermath.

MSC:

52A35 Helly-type theorems and geometric transversal theory
52-02 Research exposition (monographs, survey articles) pertaining to convex and discrete geometry
52-03 History of convex and discrete geometry
01A60 History of mathematics in the 20th century
52A37 Other problems of combinatorial convexity
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