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- Carathéodory's_theorem_(convex_hull) abstract "See also Carathéodory's theorem for other meaningsIn convex geometry Carathéodory's theorem states that if a point x of Rd lies in the convex hull of a set P, there is a subset P′ of P consisting of d + 1 or fewer points such that x lies in the convex hull of P′. Equivalently, x lies in an r-simplex with vertices in P, where . The result is named for Constantin Carathéodory, who proved the theorem in 1911 for the case when P is compact. In 1914 Ernst Steinitz expanded Carathéodory's theorem for any sets P in Rd.For example, consider a set P = {(0,0), (0,1), (1,0), (1,1)} which is a subset of R2. The convex hull of this set is a square. Consider now a point x = (1/4, 1/4), which is in the convex hull of P. We can then construct a set {(0,0),(0,1),(1,0)} = P′, the convex hull of which is a triangle and encloses x, and thus the theorem works for this instance, since |P′| = 3. It may help to visualise Carathéodory's theorem in 2 dimensions, as saying that we can construct a triangle consisting of points from P that encloses any point in P.".
- Carathéodory's_theorem_(convex_hull) thumbnail Caratheodorys_theorem_example.svg?width=300.
- Carathéodory's_theorem_(convex_hull) wikiPageID "892014".
- Carathéodory's_theorem_(convex_hull) wikiPageRevisionID "602477245".
- Carathéodory's_theorem_(convex_hull) subject Category:Articles_containing_proofs.
- Carathéodory's_theorem_(convex_hull) subject Category:Convex_hulls.
- Carathéodory's_theorem_(convex_hull) subject Category:Geometric_transversal_theory.
- Carathéodory's_theorem_(convex_hull) subject Category:Theorems_in_convex_geometry.
- Carathéodory's_theorem_(convex_hull) subject Category:Theorems_in_discrete_geometry.
- Carathéodory's_theorem_(convex_hull) comment "See also Carathéodory's theorem for other meaningsIn convex geometry Carathéodory's theorem states that if a point x of Rd lies in the convex hull of a set P, there is a subset P′ of P consisting of d + 1 or fewer points such that x lies in the convex hull of P′. Equivalently, x lies in an r-simplex with vertices in P, where . The result is named for Constantin Carathéodory, who proved the theorem in 1911 for the case when P is compact.".
- Carathéodory's_theorem_(convex_hull) label "Carathéodory's theorem (convex hull)".
- Carathéodory's_theorem_(convex_hull) label "Théorème de Carathéodory (géométrie)".
- Carathéodory's_theorem_(convex_hull) label "Теорема Каратеодори о выпуклой оболочке".
- Carathéodory's_theorem_(convex_hull) sameAs Carath%C3%A9odory's_theorem_(convex_hull).
- Carathéodory's_theorem_(convex_hull) sameAs Carathéodoryho_věta.
- Carathéodory's_theorem_(convex_hull) sameAs Théorème_de_Carathéodory_(géométrie).
- Carathéodory's_theorem_(convex_hull) sameAs Q2471737.
- Carathéodory's_theorem_(convex_hull) sameAs Q2471737.
- Carathéodory's_theorem_(convex_hull) wasDerivedFrom Carathéodory's_theorem_(convex_hull)?oldid=602477245.
- Carathéodory's_theorem_(convex_hull) depiction Caratheodorys_theorem_example.svg.
