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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All labels observed (12)

Label	Occurrences
Minkowski’s theorem	3
Minkowski convex body theorem	2
Minkowski’s convex body theorem	2
Blichfeldt theorem	1
Hermann Minkowski’s work "Geometrie der Zahlen"	1
Minkowski lattice point theorem	1
Minkowski linear forms theorem	1
Minkowski theorem	1
Minkowski theory	1
Minkowski's theorem in geometry of numbers	1
Minkowski’s first theorem	1
Minkowski’s theorem on convex sets canonical	1

How this entity was disambiguated

This entity first appeared as the object of triple T5256296 — 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: Minkowski’s theorem on convex sets
Context triple: [Carathéodory’s theorem in convex geometry, relatedTo, Minkowski’s theorem on convex sets]

A. 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.
B. Hermite–Minkowski theorem
The Hermite–Minkowski theorem is a fundamental result in algebraic number theory that gives a finiteness bound on the number of number fields of a given degree and discriminant.
C. Hilbert’s twenty-third problem
Hilbert’s twenty-third problem is one of David Hilbert’s famous list of unsolved problems, focusing on the further development and systematic application of the calculus of variations.
D. Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
E. Gauss’s remarkable theorem
Gauss’s remarkable theorem is a fundamental result in differential geometry showing that the Gaussian curvature of a surface is an intrinsic property independent of how the surface is embedded in space.
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: Minkowski’s theorem on convex sets
Target entity description: 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.

A. 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.
B. Hermite–Minkowski theorem
The Hermite–Minkowski theorem is a fundamental result in algebraic number theory that gives a finiteness bound on the number of number fields of a given degree and discriminant.
C. Hilbert’s twenty-third problem
Hilbert’s twenty-third problem is one of David Hilbert’s famous list of unsolved problems, focusing on the further development and systematic application of the calculus of variations.
D. Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
E. Gauss’s remarkable theorem
Gauss’s remarkable theorem is a fundamental result in differential geometry showing that the Gaussian curvature of a surface is an intrinsic property independent of how the surface is embedded in space.
F. None of above. chosen

Statements (46)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in the geometry of numbers ⓘ
appearsIn	Hermann Minkowski’s work "Geometrie der Zahlen" ⓘ linked to: Minkowski’s theorem on convex sets
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 ⓘ
relatedTo	Brunn–Minkowski inequality ⓘ Minkowski sum ⓘ Minkowski’s convex body theorem ⓘ linked to: Minkowski’s theorem on convex sets Minkowski’s first theorem ⓘ linked to: Minkowski’s theorem on convex sets Minkowski’s second theorem ⓘ 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 ⓘ

How these facts were elicited