Gale transform
E612749
concept in combinatorics
concept in convex geometry
mathematical construction
representation of a polytope via Gale transform
The Gale transform is a construction in convex geometry and combinatorics that represents a finite point configuration or polytope in a dual space, often used to study their structural and combinatorial properties.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Gale diagram | 1 |
| Gale transform canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6710767 — 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 transform Context triple: [David Gale, notableWork, Gale transform]
-
A.
Gauss map
The Gauss map is a differential geometry concept that assigns to each point on a surface the corresponding point on the unit sphere determined by the surface’s normal vector at that point.
-
B.
Plücker coordinates
Plücker coordinates are a system of homogeneous coordinates used in projective geometry to represent lines (and other subspaces) in higher-dimensional spaces.
-
C.
Tucker’s lemma
Tucker’s lemma is a combinatorial analog of the Borsuk–Ulam theorem that provides conditions guaranteeing the existence of certain complementary edge labels in triangulated spheres.
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D.
Sylvester–Gallai theorem
The Sylvester–Gallai theorem is a result in incidence geometry stating that for any finite set of points in the Euclidean plane not all on a single line, there exists a line that passes through exactly two of the points.
-
E.
Szekeres configuration
The Szekeres configuration is a notable geometric arrangement in projective geometry consisting of points and lines with specific incidence properties, studied for its combinatorial and symmetry characteristics.
- 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 transform Target entity description: The Gale transform is a construction in convex geometry and combinatorics that represents a finite point configuration or polytope in a dual space, often used to study their structural and combinatorial properties.
-
A.
Gauss map
The Gauss map is a differential geometry concept that assigns to each point on a surface the corresponding point on the unit sphere determined by the surface’s normal vector at that point.
-
B.
Plücker coordinates
Plücker coordinates are a system of homogeneous coordinates used in projective geometry to represent lines (and other subspaces) in higher-dimensional spaces.
-
C.
Tucker’s lemma
Tucker’s lemma is a combinatorial analog of the Borsuk–Ulam theorem that provides conditions guaranteeing the existence of certain complementary edge labels in triangulated spheres.
-
D.
Sylvester–Gallai theorem
The Sylvester–Gallai theorem is a result in incidence geometry stating that for any finite set of points in the Euclidean plane not all on a single line, there exists a line that passes through exactly two of the points.
-
E.
Szekeres configuration
The Szekeres configuration is a notable geometric arrangement in projective geometry consisting of points and lines with specific incidence properties, studied for its combinatorial and symmetry characteristics.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
concept in combinatorics
ⓘ
concept in convex geometry ⓘ mathematical construction ⓘ representation of a polytope via Gale transform ⓘ |
| appliesTo |
point configurations in Euclidean space
ⓘ
vertex sets of polytopes ⓘ |
| assumes | points are not all contained in a proper affine subspace unless dependencies are studied ⓘ |
| basedOn |
affine geometry
ⓘ
linear algebra ⓘ |
| captures |
combinatorial type of a polytope
ⓘ
incidence relations among faces of a polytope ⓘ |
| codomain | dual vector space ⓘ |
| constructionStep |
embed the affine configuration into a higher-dimensional linear space
ⓘ
start from a finite set of points in affine space ⓘ take a basis of the space of affine dependencies ⓘ use coordinates of dependency basis vectors as points in the dual space ⓘ |
| domain |
convex polytopes
ⓘ
finite point configurations ⓘ |
| field |
combinatorics
ⓘ
convex geometry ⓘ polyhedral theory ⓘ |
| hasAlternativeName |
Gale diagram
ⓘ
linked to:
Gale transform
|
| hasGeneralization | Gale duality for oriented matroids ⓘ |
| namedAfter | David Gale ⓘ |
| property |
is invariant under affine transformations of the original configuration up to linear equivalence
ⓘ
is unique up to linear isomorphism of the dual space ⓘ represents affine dependencies of original configuration as linear dependencies in the transform ⓘ |
| relatedTo |
Carathéodory theorem
ⓘ
linked to:
Carathéodory’s theorem in convex geometry
Helly theorem ⓘ
linked to:
Helly’s theorem
Radon theorem ⓘ
linked to:
Radon’s theorem
cyclic polytope ⓘ neighborly polytope ⓘ oriented matroid ⓘ |
| typicalInput | set of n points in R^d ⓘ |
| typicalOutput | set of n points in R^{n-d-1} ⓘ |
| usedFor |
analyzing combinatorial properties of polytopes
ⓘ
characterizing neighborly polytopes ⓘ detecting affine dependencies among points ⓘ studying convex polytopes ⓘ studying face lattices of polytopes ⓘ studying finite point configurations ⓘ studying projective equivalence classes of point configurations ⓘ studying realizability of oriented matroids ⓘ visualizing high-dimensional polytopes via lower-dimensional diagrams ⓘ |
| usedIn |
classification of low-dimensional polytopes
ⓘ
construction of examples and counterexamples in convex geometry ⓘ proofs of upper bound theorem for polytopes ⓘ |
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 transform Description of subject: The Gale transform is a construction in convex geometry and combinatorics that represents a finite point configuration or polytope in a dual space, often used to study their structural and combinatorial properties.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Gale transform