Minkowski’s theorem on convex sets

E506850

mathematical theorem result in the geometry of numbers

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.

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Observed surface forms (1)

Surface form	Occurrences
Minkowski convex body theorem	1

Statements (46)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in the geometry of numbers ⓘ
appearsIn	Hermann Minkowski’s work "Geometrie der Zahlen" NERFINISHED ⓘ
appliesTo	bounded convex sets ⓘ centrally symmetric convex sets ⓘ convex bodies ⓘ lattices in Euclidean space ⓘ
assumes	central symmetry of the set ⓘ convexity of the set ⓘ full-rank lattice ⓘ
conclusion	existence of a nonzero lattice point in the convex set ⓘ
coreConcept	determinant of a lattice ⓘ lattice points in convex bodies ⓘ symmetry about the origin ⓘ volume of convex sets ⓘ
domain	Euclidean space ⓘ Rn ⓘ
field	convex geometry ⓘ discrete geometry ⓘ geometry of numbers ⓘ number theory ⓘ
generalizationOf	one-dimensional pigeonhole principle for intervals and integer points ⓘ
hasApplicationIn	algebraic number theory ⓘ lattice-based cryptography (via geometry of numbers tools) ⓘ optimization and integer programming ⓘ
historicalPeriod	late 19th century ⓘ
implies	existence of nonzero lattice points in large symmetric convex bodies ⓘ
influenced	development of the geometry of numbers ⓘ lattice-based methods in number theory ⓘ modern discrete geometry ⓘ theory of convex polytopes ⓘ
namedAfter	Hermann Minkowski NERFINISHED ⓘ
relatedTo	Brunn–Minkowski inequality NERFINISHED ⓘ Minkowski sum NERFINISHED ⓘ Minkowski’s convex body theorem NERFINISHED ⓘ Minkowski’s first theorem NERFINISHED ⓘ Minkowski’s second theorem NERFINISHED ⓘ geometry of numbers ⓘ
states	Any centrally symmetric convex body in Rn with volume greater than 2n times the determinant of a lattice contains a nonzero lattice point ⓘ
typicalFormulation	If K is a centrally symmetric convex body in Rn with volume(K) > 2n det(L), then K contains a nonzero point of the lattice L ⓘ
usedFor	bounding solutions of Diophantine inequalities ⓘ lattice point enumeration problems ⓘ proving finiteness results in number theory ⓘ results on successive minima ⓘ studying convex polytopes ⓘ transference theorems in geometry of numbers ⓘ

Referenced by (2)

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

Carathéodory’s theorem in convex geometry → relatedTo → Minkowski’s theorem on convex sets ⓘ

Diophantine approximation → hasKeyResult → Minkowski’s theorem on convex sets ⓘ

this entity surface form: Minkowski convex body theorem