geometry of numbers

E637302

branch of mathematics

Geometry of numbers is a branch of number theory that studies the properties of integers and Diophantine equations using the geometry of lattices and convex bodies in Euclidean space.

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All labels observed (1)

Label	Occurrences
geometry of numbers canonical	3

How this entity was disambiguated

This entity first appeared as the object of triple T7030794 — 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: geometry of numbers
Context triple: [Diophantine approximation, relatedTo, geometry of numbers]

A. Diophantine approximation
Diophantine approximation is a branch of number theory that studies how closely real numbers can be approximated by rational numbers, often with quantitative bounds on the quality of approximation.
B. Diophantine geometry
Diophantine geometry is the branch of number theory that studies solutions to polynomial equations with integer or rational coefficients using geometric methods, particularly those from algebraic geometry.
C. Hermite constant
The Hermite constant is a number in each dimension that measures the densest possible lattice sphere packing, playing a central role in the geometry of numbers and lattice theory.
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. 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.
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: geometry of numbers
Target entity description: Geometry of numbers is a branch of number theory that studies the properties of integers and Diophantine equations using the geometry of lattices and convex bodies in Euclidean space.

A. Diophantine approximation
Diophantine approximation is a branch of number theory that studies how closely real numbers can be approximated by rational numbers, often with quantitative bounds on the quality of approximation.
B. Diophantine geometry
Diophantine geometry is the branch of number theory that studies solutions to polynomial equations with integer or rational coefficients using geometric methods, particularly those from algebraic geometry.
C. Hermite constant
The Hermite constant is a number in each dimension that measures the densest possible lattice sphere packing, playing a central role in the geometry of numbers and lattice theory.
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. 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.
F. None of above. chosen

Statements (50)

Predicate	Object
instanceOf	branch of mathematics ⓘ
appliesTo	Diophantine approximation ⓘ Diophantine inequalities ⓘ algebraic number theory ⓘ lattice point counting problems ⓘ quadratic forms ⓘ sphere packing problems ⓘ transference principles ⓘ
centralConcept	Minkowski sum ⓘ convex symmetric body ⓘ covering radius ⓘ lattice in Euclidean space ⓘ packing density ⓘ reduction theory of quadratic forms ⓘ successive minima ⓘ
developedIn	early 20th century ⓘ late 19th century ⓘ
field	number theory ⓘ
hasApplicationIn	coding theory ⓘ cryptography ⓘ discrete tomography ⓘ optimization ⓘ
hasMethod	lattice point enumeration in convex bodies ⓘ reduction of lattices ⓘ successive minima estimates ⓘ volume comparison arguments ⓘ
hasTheorem	Blichfeldt theorem ⓘ linked to: Minkowski’s theorem on convex sets Hermite constant bounds ⓘ linked to: Hermite constant Mahler compactness theorem ⓘ Minkowski convex body theorem ⓘ linked to: Minkowski’s theorem on convex sets Minkowski lattice point theorem ⓘ linked to: Minkowski’s theorem on convex sets Minkowski linear forms theorem ⓘ linked to: Minkowski’s theorem on convex sets Siegel mean value theorem ⓘ
historicalFigure	Carl Ludwig Siegel ⓘ Hermann Minkowski ⓘ Kurt Mahler ⓘ Louis Mordell ⓘ
introducedBy	Hermann Minkowski ⓘ
relatedTo	algebraic geometry ⓘ arithmetic geometry ⓘ discrete geometry ⓘ functional analysis ⓘ metric number theory ⓘ
studies	Diophantine equations ⓘ convex bodies ⓘ integer points in Euclidean space ⓘ lattices ⓘ
uses	Euclidean geometry ⓘ convex geometry ⓘ lattice theory ⓘ

How these facts were elicited

Referenced by (3)

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

Diophantine approximation → relatedTo → geometry of numbers ⓘ

Hermite–Minkowski theorem → relatedTo → geometry of numbers ⓘ

Dirichlet approximation theorem → usedIn → geometry of numbers ⓘ

All labels observed (1)

How this entity was disambiguated Show

Statements (50)

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Referenced by (3)

How this entity was disambiguated

How these facts were elicited