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 (1)
| Label | Occurrences |
|---|---|
| Gale’s theorem on linear inequalities canonical | 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]
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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.
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.
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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.
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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.
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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 NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| mathematicianAssociated | David Gale NERFINISHED ⓘ |
| namedAfter | David Gale NERFINISHED ⓘ |
| provides | alternative system of linear inequalities ⓘ |
| relatedTo |
Farkas’ lemma
NERFINISHED
ⓘ
Gordan’s theorem NERFINISHED ⓘ Hahn–Banach separation theorem NERFINISHED ⓘ Stiemke’s theorem NERFINISHED ⓘ 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
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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 (1)
Full triples — surface form annotated when it differs from this entity's canonical label.