Helly's Theorem states that for a finite collection of convex sets in d-dimensional space, if every subcollection of d plus one of the sets has a point in common, then the entire collection has a point in common. Named for Eduard Helly, it is a foundational result of convex geometry that reduces checking a common intersection for arbitrarily many convex sets to checking only small subcollections of a bounded size, and it underlies later results such as Radon's Theorem and the centerpoint theorem.
Facts
StatementFor a finite collection of convex subsets of R^d with n >= d+1, if the intersection of every d+1 of them is nonempty, then the whole collection has a nonempty intersection. 1 Classification
Statement Form Helly's Theorem
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Source Helly's theorem and related results (Wikipedia)
Principal Result Of
Source MacTutor History of Mathematics, University of St Andrews: Eduard Helly
Proved By
Source Helly's theorem and related results (Wikipedia)
Sources
1. Helly's theorem and related results (Wikipedia)
Proof
In this new collection, every subcollection of d + 1 sets will have nonempty intersection.
Introduction, discovery
It was discovered by Eduard Helly in 1913, but not published by him until 1923, by which time alternative proofs by Radon (1921) and König (1922) had already appeared.
In Branch: Discrete Geometry, Lead sentence
Helly's theorem is a basic result in discrete geometry on the intersection of convex sets.
Proved By: Eduard Helly, Lead sentence, attribution
discovered by Eduard Helly in 1913
View the SourceMacTutor History of Mathematics, University of St Andrews: Eduard Helly
MacTutor History of Mathematics, University of St AndrewsPrincipal Result Of: Eduard Helly, Biography textQuote, Principal Result Of: Eduard Helly, Biography text
He is remembered for Helly's theorem, published in 1923 , which states that if there are given n n n convex subsets of a d d d -dimensional Euclidean space with n ≥ d + 1 n ≥ d+1 n ≥ d + 1 and if each collection of d + 1 d + 1 d + 1 of the subsets has a point in common then there is a common point o
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