Convex Polytopes and Polyhedra
- Introduction and First Examples
- Polytopes, Cones, and Polyhedra
- Polytopes
Polytope(Q): SeqEnum → TorPolPolyhedronWithInequalities(A,c): SeqEnum, [RngIntElt] → TorPolPolyhedronWithInequalities(A,c): [TorLatElt], [RngIntElt] → TorPolRandomPolytope(L,n,k): TorLat, RngIntElt, RngIntElt → TorPolRandomPolytope(d,n,k): RngIntElt, RngIntElt, RngIntElt → TorPolBoundingBox(P): TorPol → TorPol, TorLatElt, TorLatEltPolar(P): TorPol → TorPolCrossPolytope(L): TorLat → TorPolCrossPolytope(d): RngIntElt → TorPolStandardSimplex(L): TorLat → TorPolStandardSimplex(d): RngIntElt → TorPolCyclicPolytope(L,n): TorLat, RngIntElt → TorPolCyclicPolytope(d,n): RngIntElt, RngIntElt → TorPolPolytopeOfProjectiveSpace(d): RngIntElt → TorPolPolytopeOfProjectiveSpace(L): TorLat → TorPolPolytopeOfWPS(d): RngIntElt → TorPolPolytopeOfWPS(W): [RngIntElt] → TorPolPolytopeOfWPS(L,W): [RngIntElt] → TorPol
- Cones
Cone(A): Seq → TorConCone(v): TorLatElt → TorConConeWithInequalities(B): Set → TorConFullCone(L): TorLat → TorConFullCone(n): RngIntElt → TorConPositiveQuadrant(L): TorLat → TorConPositiveQuadrant(n): RngIntElt → TorConZeroCone(L): TorLat → TorConZeroCone(n): RngIntElt → TorConRandomCone(d,n,k): RngIntElt, RngIntElt, RngIntElt → TorConRandomCone(L,n,k): TorLat, RngIntElt, RngIntElt → TorConRandomPositiveCone(d,n,k): RngIntElt, RngIntElt, RngIntElt → TorConRandomPositiveCone(L,n,k): TorLat, RngIntElt, RngIntElt → TorConDual(C): TorCon → TorConNormalisedCone(P): TorPol → TorConConeInSublattice(C): TorCon → TorCon, MapConeQuotientByLinearSubspace(C): TorCon → TorCon, Map, MapSimplicialSubcone(C): TorCon → TorConLatticeBasisInCone(C): TorCon → [TorLatElt]
- Polyhedra
Polyhedron(C,H,h): TorCon, TorLatElt, FldRatElt → TorPolPolyhedron(C,H,h): TorCon, TorLatElt, RngIntElt → TorPolPolyhedron(C): TorCon → TorPolHalfspaceToPolyhedron(v,h): TorLatElt, FldRatElt → TorPolHalfspaceToPolyhedron(Q,h): [FldRatElt], FldRatElt → TorPolHyperplaneToPolyhedron(v,h): TorLatElt, FldRatElt → TorPolHyperplaneToPolyhedron(Q,h): [FldRatElt], FldRatElt → TorPolPolyhedron(C,f,v): TorCon, Map, TorLatElt → TorPolEmptyPolyhedron(L): TorLat → TorPolConeToPolyhedron(C): TorCon → TorPolPolyhedronInSublattice(P): TorPol → TorPol, Map, TorLatEltFixedSubspaceToPolyhedron(G): GrpMat → TorPolFixedSubspaceToPolyhedron(L,G): TorLat, GrpMat → TorPol- Example: Polyhedra as Slices of a Quadrant
- Arithmetic Operations on Polyhedra
C eq D: TorCon, TorCon → BoolEltP eq Q: TorPol, TorPol → BoolEltC meet D: TorCon, TorCon → TorConP meet Q: TorPol, TorPol → TorPolP subset Q: TorPol, TorPol → BoolEltC + D: TorCon, TorCon → TorConP + Q: TorPol, TorPol → TorPolP + C: TorPol, TorCon → TorPolC + P: TorCon, TorPol → TorPolP * Q: TorPol, TorPol → TorPolk * P: FldRatElt, TorPol → TorPol- P: TorPol → TorPol
- Basic Combinatorics of Polytopes and Polyhedra
- Vertices and Inequalities
- Facets and Faces
fVector(C): TorCon → SeqEnum[RngIntElt]fVector(P): TorPol → SeqEnum[RngIntElt]hVector(C): TorCon → SeqEnum[RngIntElt]hVector(P): TorPol → SeqEnum[RngIntElt]Facets(C): TorCon → SeqEnumFacets(P): TorPol → SeqEnumFacetIndices(P): TorPol → SeqEnumNumberOfFacets(P): TorPol → RngIntEltFaces(C): TorCon → SeqEnumFaces(P): TorPol → SeqEnumFaces(C,i): TorCon, RngIntElt → SeqEnumFaces(P,i): TorPol, RngIntElt → SeqEnumFaceIndices(P,i): TorPol, RngIntElt → SeqEnumNumberOfFaces(P,i): TorPol, RngIntElt → RngIntEltEdges(P): TorPol → SeqEnumEdgeIndices(P): TorPol → SeqEnumNumberOfEdges(P): TorPol → RngIntEltGraph(P): TorPol → GrphUndFaceSupportedBy(C,H): TorCon, TorLatElt → TorConIsSupportingHyperplane(v,h,P): TorLatElt, FldRatElt, TorPol → BoolElt, RngIntEltSupportingCone(P,v): TorPol, TorLatElt → TorConIsFace(C,F): TorCon, TorCon → BoolEltIsFace(P,F): TorPol, TorPol → BoolElt
- The Combinatorics of Polytopes
- Points in Polytopes and Polyhedra
v in C: TorLatElt, TorCon → BoolEltv in P: TorLatElt, TorPol → BoolEltIsInInterior(v,C): TorLatElt, TorCon → BoolEltIsInInterior(v,P): TorLatElt, TorPol → BoolEltIsOnBoundary(v,C): TorLatElt, TorCon → BoolEltIsOnBoundary(v,P): TorLatElt, TorPol → BoolEltHasIntegralPoint(P): TorPol → BoolEltPoints(P): TorPol → SeqEnum[TorLatElt]InteriorPoints(P): TorPol → SeqEnum[TorLatElt]BoundaryPoints(P): TorPol → SeqEnum[TorLatElt]NumberOfPoints(P): TorPol → RngIntEltNumberOfInteriorPoints(P): TorPol → RngIntEltNumberOfBoundaryPoints(P): TorPol → RngIntEltVolume(P): TorPol → FldRatEltVolumeOfBoundary(P): TorPol → FldRatElt
- Ehrhart Theory of Polytopes
- Isomorphism Testing and Normal Forms for Polytopes
IsIsomorphic(P,Q): TorPol, TorPol → BoolElt, Map- Example: Isomorphism Testing for Polytopes
IsEquivalent(P,Q): TorPol, TorPol → BoolElt, Map, TorLatEltNormalForm(P): TorPol → SeqEnum, GrpPermEltPALPNormalForm(P): TorPol → SeqEnum, GrpPermElt- Example: Normal Form of a Lattice Polytope
AffineNormalForm(P): TorPol → SeqEnum, GrpPermElt- Example: Classifying Polygons by Affine Normal Form
MaximalVertexFacetHeightMatrix(P): TorPol → AlgMatElt
- Automorphisms of a Polytope
- Operations on Polytopes
- Cones and Polyhedra
- Generators of Cones
- Properties of Polyhedra
- Attributes of Polyhedra
IsPolytope(P): TorPol → BoolEltDimension(C): TorCon → RngIntEltDimension(P): TorPol → RngIntEltDegree(P): TorPol → RngIntEltCodegree(P): TorPol → RngIntEltIndex(C): TorCon → RngIntEltWidth(P): TorPol → FldRatElt, SetEnumWidth(P,u): TorPol, TorLatElt → FldRatElt- Example: Lattice Width of a Polytope
IsPyramid(P): TorPol → BoolElt, TorLatElt, TorPol, Map, TorLatEltPyramid(P): TorPol → TorPol, Map, Map, Map, Map- Example: Recognising a Pyramid
VertexEdgeIncidenceMatrix(P): TorPol → ModMatRngEltVertexFacetIncidenceMatrix(P): TorPol → ModMatRngEltVertexFacetHeightMatrix(P): TorPol → AlgMatEltEdgeFacetIncidenceMatrix(P): TorPol → ModMatRngElt
- Combinatorics of Polyhedral Complexes
- Toric Lattices
- Toric Lattices
ToricLattice(n): RngIntElt → TorLatScalarLattice() → TorLat- Example: Summing an Empty Sequence of Points
Dual(L): TorLat → TorLat- Example: Dual Toric Lattice
L + M: TorLat, TorLat → TorLat, TorLatMap, TorLatMap, TorLatMap, TorLatMapDirectSum(L,M): TorLat, TorLat → TorLat, TorLatMap, TorLatMap, TorLatMap, TorLatMapDirectSum(Q): SeqEnum → TorLat, TorLatMap, TorLatMap, TorLatMap, TorLatMapL ^ n: TorLat, RngIntElt → TorLat, SeqEnum, SeqEnumDimension(L): TorLat → RngIntElt
- Points of Toric Lattices
L ! [a,b,...]: TorLat, [RngIntElt] → TorLatEltLatticeVector(L,Q): TorLat, [RngIntElt] → TorLatEltLatticeVector(Q): [RngIntElt] → TorLatEltL . i: TorLat, RngIntElt → TorLatEltBasis(L,i): TorLat, RngIntElt → TorLatEltBasis(L): TorLat → TorLatEltForm(L,Q): TorLat, [RngIntElt] → TorLatEltZero(L): TorLat → TorLatEltP + Q: TorLatElt, TorLatElt → TorLatEltP - Q: TorLatElt, TorLatElt → TorLatEltn * P: FldRatElt, TorLatElt → TorLatEltP / n: TorLatElt, FldRatElt → TorLatEltP eq Q: TorLatElt, TorLatElt → BoolEltAreProportional(P,Q): TorLatElt, TorLatElt → BoolElt, FldRatEltP / Q: TorLatElt, TorLatElt → FldRatElt- Example: Arithmetic with Toric Lattice Points
v in L: TorLatElt, TorLat → BoolEltMatrix(R,S): Rng, [TorLatElt] → ModMatRngEltMatrix(S): [TorLatElt] → ModMatRngEltVector(v): TorLatElt → ModTupFldEltIsZero(v): TorLatElt → BoolEltIsIntegral(v): TorLatElt → BoolEltIsPrimitive(v): TorLatElt → BoolEltPrimitiveLatticeVector(v): TorLatElt → TorLatElt- Example: Primitive Points in a Sublattice
- Operations on Toric Lattices
L eq K: TorLat, TorLat → BoolEltSublattice(Q): [TorLatElt] → TorLat, TorLatMapToricLattice(Q): [[RngIntElt]] → TorLat, TorLatMapQuotient(C): TorCon → TorLat, TorLatMapQuotient(Q): [TorLatElt] → TorLat, TorLatMapQuotient(v): TorLatElt → TorLat, TorLatMapAddVectorToLattice(v): TorLatElt → TorLat, TorLatMapAddVectorToLattice(Q): [TorLatElt] → TorLat, TorLatMapAreGenerators(S): SetEnum → BoolEltIsSublattice(L): TorLat → BoolEltIsSuperlattice(L): TorLat → BoolEltIsDirectSum(L): TorLat → BoolEltIsQuotient(L): TorLat → BoolEltSublattice(L): TorLat → TorLat, TorLatMapSuperlattice(L): TorLat → TorLat, TorLatMapSummands(L): TorLat → SeqEnum, SeqEnum, SeqEnum- Example: Basis of a Toric Sublattice
- Maps of Toric Lattices
ZeroMap(L,K): TorLat, TorLat → TorLatMapIdentityMap(L): TorLat → TorLatMaphom< L -> K | M >: TorLat, TorLat, Mtrx → TorLatMapLatticeMap(L,K,M): TorLat, TorLat, Mtrx → TorLatMapLatticeMap(L,Q): TorLat, [TorLatElt] → TorLatMapDefiningMatrix(f): TorLatMap → ModMatRngEltImage(f,C): TorLatMap, TorCon → TorConImage(f,P): TorLatMap, TorPol → TorPolImage(f,v): TorLatMap, TorLatElt → TorLatEltPreimage(f,C): TorLatMap, TorCon → TorConPreimage(f,P): TorLatMap, TorPol → TorPolPreimage(f,v): TorLatMap, TorLatElt → TorLatEltKernelEmbedding(f): TorLatMap → MapKernelEmbedding(v): TorLatElt → MapKernelBasis(f): TorLatMap → SeqEnumKernelBasis(v): TorLatElt → SeqEnumImageBasis(f): TorLatMap → SeqEnumIsCokernelTorsionFree(f): TorLatMap → BoolEltChangeBasis(v): TorLatElt → Map- Example: Change of Basis Revealing Lattice Width
- Bibliography