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Helly’s theorem: new variations and applications. (English) Zbl 1383.52006

Harrington, Heather A. (ed.) et al., Algebraic and geometric methods in discrete mathematics. AMS special session on algebraic and geometric methods in applied discrete mathematics, San Antonio, TX, USA, January 11, 2015. Proceedings. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-2321-6/pbk; 978-1-4704-3743-5/ebook). Contemporary Mathematics 685, 55-95 (2017).
This paper is an extensive survey on the Helly theorem and related results. The Helly theorem is a classical result in discrete geometry and it states the following: if we are given a finite collection of at least \(d+1\) convex sets in \(\mathbb{R}^d\) such that the intersection of every \((d+1)\)-tuple of sets in the collection is nonempty, then the intersection of all sets in the collection is nonempty. This innocent-looking theorem has a plethora of generalizations, variants, or counterparts, as well as interesting applications (for example in optimization).
The task of the survey is to describe the (mostly) recent development of the results related to Helly theorem, mostly considering the geometric setting. In particular, the survey includes (quite exhaustive) description of the following topics.
\(\bullet\)
Helly-type theorems where the intersection is considered in a prescribed set.
\(\bullet\)
Colorful versions. (The sets are colored and the intersection condition is related to the coloring.)
\(\bullet\)
Quantitative versions. (The intersection is supposed to be large in some sense.)
\(\bullet\)
Topological versions. (The convexity condition is relaxed to a weaker topological condition.)
\(\bullet\)
Fractional versions. (Only a fraction of intersections is assumed to be nonempty with conclusion that a fraction of the sets has nonempty intersection.)
\(\bullet\)
\((p,q)\) versions. (For \(p \geq q\), if among every \(p\) sets there are \(q\) sets with non-empty intersections, deduce that all sets can be hit by a bounded number of points.)
\(\bullet\)
Transversal theorems. (Nonempty intersection is replaced with existence of transversal line, affine space, etc.)
\(\bullet\)
Applications of Helly-type theorems in optimization (and vice versa).

For the entire collection see [Zbl 1362.00040].

MSC:

52A35 Helly-type theorems and geometric transversal theory
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References:

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