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.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Gale diagram | 0 |
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 NERFINISHED ⓘ |
| hasGeneralization | Gale duality for oriented matroids ⓘ |
| namedAfter | David Gale NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
Helly theorem NERFINISHED ⓘ Radon theorem NERFINISHED ⓘ 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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.