Blaschke selection theorem

E853120

compactness theorem mathematical theorem result in convex geometry

The Blaschke selection theorem is a fundamental result in convex geometry and functional analysis that guarantees the existence of a convergent subsequence in any bounded sequence of convex bodies under the Hausdorff metric.

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All labels observed (1)

Label	Occurrences
Blaschke selection theorem canonical	2

How this entity was disambiguated

This entity first appeared as the object of triple T10269882 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Blaschke selection theorem
Context triple: [Wilhelm Blaschke, knownFor, Blaschke selection theorem]

A. Helly’s theorem
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.
B. Krein–Milman theorem
The Krein–Milman theorem is a fundamental result in functional analysis and convex geometry stating that a compact convex set in a locally convex topological vector space is the closed convex hull of its extreme points.
C. Carathéodory’s theorem in convex geometry
Carathéodory’s theorem in convex geometry is a fundamental result stating that any point in the convex hull of a set in ℝⁿ can be expressed as a convex combination of at most n+1 points from that set.
D. Minkowski’s theorem on convex sets
Minkowski’s theorem on convex sets is a fundamental result in convex geometry that characterizes lattice points in convex bodies, underpinning much of the theory of convex polytopes and the geometry of numbers.
E. Banach–Mazur theorem
The Banach–Mazur theorem is a fundamental result in functional analysis that characterizes separable Banach spaces as isometrically isomorphic to closed subspaces of spaces of continuous functions on compact metric spaces.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Blaschke selection theorem
Target entity description: The Blaschke selection theorem is a fundamental result in convex geometry and functional analysis that guarantees the existence of a convergent subsequence in any bounded sequence of convex bodies under the Hausdorff metric.

A. Helly’s theorem
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.
B. Krein–Milman theorem
The Krein–Milman theorem is a fundamental result in functional analysis and convex geometry stating that a compact convex set in a locally convex topological vector space is the closed convex hull of its extreme points.
C. Carathéodory’s theorem in convex geometry
Carathéodory’s theorem in convex geometry is a fundamental result stating that any point in the convex hull of a set in ℝⁿ can be expressed as a convex combination of at most n+1 points from that set.
D. Minkowski’s theorem on convex sets
Minkowski’s theorem on convex sets is a fundamental result in convex geometry that characterizes lattice points in convex bodies, underpinning much of the theory of convex polytopes and the geometry of numbers.
E. Banach–Mazur theorem
The Banach–Mazur theorem is a fundamental result in functional analysis that characterizes separable Banach spaces as isometrically isomorphic to closed subspaces of spaces of continuous functions on compact metric spaces.
F. None of above. chosen

Statements (42)

Predicate	Object
instanceOf	compactness theorem ⓘ mathematical theorem ⓘ result in convex geometry ⓘ
appearsIn	classical textbooks on convex geometry ⓘ monographs on geometric functional analysis ⓘ
appliesTo	bounded sequences of convex bodies ⓘ closed convex subsets of R^n ⓘ convex bodies in Euclidean space ⓘ
assumes	boundedness in the Hausdorff metric ⓘ nonempty convex compact sets ⓘ
concerns	Hausdorff metric ⓘ compactness of families of convex sets ⓘ convex bodies ⓘ
context	space of convex bodies endowed with Hausdorff metric ⓘ
field	convex geometry ⓘ functional analysis ⓘ geometric measure theory ⓘ
generalizationOf	compactness of closed intervals in R ⓘ
guarantees	existence of a convergent subsequence ⓘ relative compactness in the Hausdorff metric ⓘ sequential compactness of bounded families of convex bodies ⓘ
historicalPeriod	early 20th century ⓘ
holdsIn	finite-dimensional Euclidean spaces ⓘ
implies	compactness of the space of convex bodies modulo translations under Hausdorff metric ⓘ existence of limit shapes for bounded sequences of convex bodies ⓘ
involves	convergence of sets ⓘ metric topology on sets ⓘ
namedAfter	Wilhelm Blaschke ⓘ
relatedTo	Banach–Alaoglu theorem ⓘ Carathéodory's theorem ⓘ linked to: Carathéodory’s theorem in convex geometry Helly's theorem ⓘ linked to: Helly’s theorem Krein–Milman theorem ⓘ Prokhorov's theorem ⓘ linked to: Prokhorov’s theorem
typicalFormulation	Every bounded sequence of convex bodies in R^n has a subsequence converging in the Hausdorff metric to a convex body ⓘ
usedIn	Minkowski addition theory ⓘ linked to: Minkowski sum asymptotic convex geometry ⓘ geometric functional analysis ⓘ isoperimetric problems ⓘ shape optimization ⓘ theory of random polytopes ⓘ
uses	Hausdorff distance ⓘ linked to: Hausdorff metric compactness arguments ⓘ

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Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Wilhelm Blaschke → knownFor → Blaschke selection theorem ⓘ

Wilhelm Blaschke → hasConceptNamedAfter → Blaschke selection theorem ⓘ

subject linked to: Blaschke

All labels observed (1)

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Statements (42)

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Referenced by (2)

How this entity was disambiguated

How these facts were elicited