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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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 NERFINISHED ⓘ 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 NERFINISHED ⓘ
relatedTo	Banach–Alaoglu theorem NERFINISHED ⓘ Carathéodory's theorem NERFINISHED ⓘ Helly's theorem NERFINISHED ⓘ Krein–Milman theorem NERFINISHED ⓘ Prokhorov's theorem NERFINISHED ⓘ
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 NERFINISHED ⓘ asymptotic convex geometry ⓘ geometric functional analysis ⓘ isoperimetric problems ⓘ shape optimization ⓘ theory of random polytopes ⓘ
uses	Hausdorff distance NERFINISHED ⓘ compactness arguments ⓘ

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Wilhelm Blaschke → knownFor → Blaschke selection theorem ⓘ