Helly's theorem


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Helly's theorem

[′hel·ēz ‚thir·əm]
(mathematics)
The theorem that there is a point that belongs to each member of a collection of bounded closed convex sets in an n-dimensional Euclidean space if the collection has at least n + 1 members and any n + 1 members of the collection have a common point.
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It describes the linear or vector space concepts of addition and scalar multiplication, linear subspaces, linear functionals, and hyperplanes, as well as different distances in n-space and the geometric properties of subsets, subspaces, and hyperplanes; topology in the context of metrics derived from a norm on the n-dimensional space; the concept of convexity and the basic properties of convex sets; and Helly's theorem and applications involving transversals of families of pairwise disjoint compact convex subsets of the plane.
The Centerpoint Theorem, which is a simple consequence of Helly's Theorem [6], states that for any point set S of size n there exists a point whose halfspace depth is at least [n/(d + 1)].
By Helly's Theorem, there exists a weak-star limit [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] along some subsequence [LAMBDA] [subset] N.