Minkowski’s theorem on convex sets
E506850
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Minkowski convex body 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]
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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.
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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.
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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.
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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.
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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" 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 ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Minkowski’s theorem on convex sets Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Minkowski convex body theorem