Gale’s theorem on linear inequalities
E612748
Gale’s theorem on linear inequalities is a fundamental result in convex geometry and linear programming that characterizes the solvability of systems of linear inequalities via an associated alternative system.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Farkas’ lemma | 1 |
| Gale’s theorem of alternatives | 1 |
| Gale’s theorem on linear inequalities canonical | 1 |
| Stiemke’s theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6710766 — 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: Gale’s theorem on linear inequalities Context triple: [David Gale, notableWork, Gale’s theorem on linear inequalities]
-
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.
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.
-
C.
Helly’s theorem
Helly’s theorem is a fundamental result in convex geometry that gives conditions under which a family of convex sets in Euclidean space has a nonempty common intersection.
-
D.
Grothendieck inequality
The Grothendieck inequality is a fundamental result in functional analysis and theoretical computer science that bounds certain bilinear forms and has deep implications for Banach space theory, operator theory, and approximation algorithms.
-
E.
Radon’s theorem
Radon’s theorem is a fundamental result in convex geometry stating that any set of sufficiently many points in Euclidean space can be partitioned into two disjoint subsets whose convex hulls intersect.
- 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: Gale’s theorem on linear inequalities Target entity description: Gale’s theorem on linear inequalities is a fundamental result in convex geometry and linear programming that characterizes the solvability of systems of linear inequalities via an associated alternative system.
-
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.
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.
-
C.
Helly’s theorem
Helly’s theorem is a fundamental result in convex geometry that gives conditions under which a family of convex sets in Euclidean space has a nonempty common intersection.
-
D.
Grothendieck inequality
The Grothendieck inequality is a fundamental result in functional analysis and theoretical computer science that bounds certain bilinear forms and has deep implications for Banach space theory, operator theory, and approximation algorithms.
-
E.
Radon’s theorem
Radon’s theorem is a fundamental result in convex geometry stating that any set of sufficiently many points in Euclidean space can be partitioned into two disjoint subsets whose convex hulls intersect.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in convex geometry ⓘ result in linear programming ⓘ |
| alsoKnownAs |
Gale’s theorem of alternatives
ⓘ
linked to:
Gale’s theorem on linear inequalities
|
| appliesTo |
feasibility problems in linear programming
ⓘ
systems of linear inequalities ⓘ |
| category |
theorem in functional analysis
ⓘ
theorem in real analysis ⓘ |
| characterizes | solvability of systems of linear inequalities ⓘ |
| concerns |
existence of solutions to linear inequalities
ⓘ
infeasibility certificates for linear systems ⓘ |
| field |
convex geometry
ⓘ
linear programming ⓘ optimization theory ⓘ |
| framework | finite-dimensional real vector spaces ⓘ |
| gives | necessary and sufficient conditions for solvability of linear inequalities ⓘ |
| hasConcept |
alternative system
ⓘ
certificate of infeasibility ⓘ primal–dual relationship ⓘ |
| implies | mutual exclusivity of primal and alternative systems ⓘ |
| influenced | modern formulations of linear programming duality ⓘ |
| involves |
convex cones
ⓘ
homogeneous linear inequalities ⓘ nonhomogeneous linear inequalities ⓘ separation properties of convex sets ⓘ |
| isPartOf |
classical theory of linear inequalities
ⓘ
foundations of mathematical economics ⓘ |
| logicalForm | either-or alternative theorem ⓘ |
| mathematicianAssociated | David Gale ⓘ |
| namedAfter | David Gale ⓘ |
| provides | alternative system of linear inequalities ⓘ |
| relatedTo |
Farkas’ lemma
ⓘ
linked to:
Gale’s theorem on linear inequalities
Gordan’s theorem ⓘ Hahn–Banach separation theorem ⓘ
linked to:
Hahn–Banach theorem
Stiemke’s theorem ⓘ
linked to:
Gale’s theorem on linear inequalities
convex separation theorems ⓘ theorems of the alternative ⓘ |
| typeOf |
duality statement
ⓘ
separation theorem ⓘ |
| usedAs |
tool for analyzing feasibility regions
ⓘ
tool for proving existence of equilibria ⓘ |
| usedIn |
duality theory in linear programming
ⓘ
economic equilibrium theory ⓘ feasibility analysis of linear systems ⓘ game theory ⓘ proofs of strong duality in linear programming ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Gale’s theorem on linear inequalities Description of subject: Gale’s theorem on linear inequalities is a fundamental result in convex geometry and linear programming that characterizes the solvability of systems of linear inequalities via an associated alternative system.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Gale’s theorem on linear inequalities
linked to: Gale’s theorem on linear inequalities
linked to: Gale’s theorem on linear inequalities