Statements (69)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:mathematical_concept
|
| gptkbp:application |
gptkb:Computer_graphics
gptkb:Game_theory gptkb:Computational_geometry mathematical optimization Operations research |
| gptkbp:characterizedBy |
Convex hull of a finite set of points
Finite intersection of half-spaces |
| gptkbp:definedIn |
A polytope that is also a convex set
|
| gptkbp:dimensions |
n-dimensional
|
| gptkbp:example |
Convex polyhedron
Convex polygon |
| gptkbp:faced |
gptkb:Convex_polytope
|
| gptkbp:field |
gptkb:Mathematics
gptkb:geometry Combinatorics |
| gptkbp:hasFacet |
gptkb:Convex_polytope
|
| gptkbp:hasSpecialCase |
gptkb:Polytopes
|
| gptkbp:hasVertex |
Extreme point
|
| https://www.w3.org/2000/01/rdf-schema#label |
Convex Polytopes
|
| gptkbp:notablePerson |
gptkb:Branko_Grünbaum
gptkb:Hermann_Minkowski gptkb:Ludwig_Schläfli gptkb:Eugène_Charles_Catalan |
| gptkbp:notableWork |
Grünbaum's Convex Polytopes
|
| gptkbp:property |
Vertices, edges, faces, and higher-dimensional analogues
Minkowski sum of convex polytopes is a convex polytope Bounded convex polyhedron in 3D Can be centrally symmetric Can be circumscribed about a sphere Can be described by convex hull of points Can be described by intersection of half-spaces Can be inscribed in a sphere Can be regular, semi-regular, or irregular Can be simple or simplicial Carathéodory's theorem applies Combinatorial type determined by face lattice Dehn–Sommerville equations apply Described by V-representation and H-representation Dual polytope exists Euler characteristic applies in 3D Face lattice structure Faces are themselves convex polytopes Finite number of faces Gale diagram representation Helly's theorem applies Krein–Milman theorem applies McMullen's Upper Bound Theorem applies Neighborliness property Radon's theorem applies Shellability property Triangulation possible Volume and surface area are well-defined f-vector describes face numbers h-vector describes combinatorial structure Projection of a convex polytope is a convex polytope Can be realized as solution sets of linear inequalities Intersection of convex polytopes is a convex polytope Every line segment between two points in the polytope lies entirely within the polytope |
| gptkbp:relatedTo |
gptkb:Cross-polytope
gptkb:Polyhedral_combinatorics gptkb:Simplex gptkb:Cube Convex hull |
| gptkbp:studiedBy |
gptkb:Euclidean_geometry
|
| gptkbp:studiedIn |
Linear programming
Optimization |
| gptkbp:bfsParent |
gptkb:Branko_Grünbaum
|
| gptkbp:bfsLayer |
6
|