Helly’s theorem

E506847

mathematical theorem result in convex geometry

Helly’s theorem is a fundamental result in convex geometry that gives conditions under which a family of convex sets in Euclidean space has a nonempty common intersection.

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Label	Occurrences
Helly’s theorem canonical	1

Statements (47)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in convex geometry ⓘ
appearsIn	classical convexity theory ⓘ
appliesIn	Euclidean space NERFINISHED ⓘ
appliesTo	families of convex sets ⓘ
assertsThat	for a finite family of convex sets in R^d, if every subfamily of size d+1 has nonempty intersection, then the whole family has nonempty intersection ⓘ
category	theorems in convex analysis ⓘ theorems in geometry ⓘ
coreConcept	intersection of convex sets ⓘ
dimensionDependent	yes ⓘ
field	combinatorial geometry ⓘ convex geometry ⓘ discrete geometry ⓘ
givesConditionFor	nonempty common intersection ⓘ
hasGeneralization	Helly-type theorems for algebraic sets NERFINISHED ⓘ Helly-type theorems for other set systems ⓘ Helly-type theorems in metric spaces ⓘ
hasHellyNumber	d+1 for convex sets in R^d ⓘ
hasParameter	dimension d of Euclidean space ⓘ
hasVariant	Doignon’s theorem NERFINISHED ⓘ quantitative Helly theorem NERFINISHED ⓘ topological Helly theorem NERFINISHED ⓘ
holdsFor	finite families of convex sets ⓘ
implies	finite intersection property for convex sets under Helly’s condition ⓘ
influenced	development of combinatorial convexity ⓘ theory of LP-type problems ⓘ
isFiniteVersionOf	intersection properties of convex sets ⓘ
isToolFor	geometric proofs in combinatorics ⓘ proving existence of feasible solutions in linear inequalities ⓘ
namedAfter	Eduard Helly NERFINISHED ⓘ
originallyProvedBy	Eduard Helly NERFINISHED ⓘ
publicationLanguage	German ⓘ
relatedTo	(p,q)-theorem NERFINISHED ⓘ Carathéodory’s theorem NERFINISHED ⓘ Radon’s theorem NERFINISHED ⓘ Tverberg’s theorem NERFINISHED ⓘ colorful Helly theorem NERFINISHED ⓘ fractional Helly theorem NERFINISHED ⓘ
specialCaseOf	Helly-type theorems NERFINISHED ⓘ
usedIn	computational geometry ⓘ discrete optimization ⓘ geometric algorithms ⓘ geometric transversal theory ⓘ linear programming theory ⓘ optimization ⓘ
yearProvedApprox	1913 ⓘ
yearPublishedApprox	1923 ⓘ

Referenced by (1)

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Carathéodory’s theorem in convex geometry → relatedTo → Helly’s theorem ⓘ