Carathéodory's theorem
Encyclopedia
In mathematics
, Carathéodory's theorem may refer to one of a number of results of Constantin Carathéodory
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Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Carathéodory's theorem may refer to one of a number of results of Constantin Carathéodory
Constantin Carathéodory
Constantin Carathéodory was a Greek mathematician. He made significant contributions to the theory of functions of a real variable, the calculus of variations, and measure theory...
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- Carathéodory's theorem (conformal mapping)Carathéodory's theorem (conformal mapping)In mathematical complex analysis, Carathéodory's theorem, proved by , states that if U is a simply connected open subset of the complex plane C, whose boundary is a Jordan curve Γ then the Riemann map...
, about the extension of conformal mappings to the boundary - Carathéodory's theorem (convex hull)Carathéodory's theorem (convex hull)In 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 r \leq d...
, about the convex hulls of sets in Euclidean space - Carathéodory's theorem (measure theory), about outer measures in measure theory
- Carathéodory's existence theoremCarathéodory's existence theoremIn mathematics, Carathéodory's existence theorem says that an ordinary differential equation has a solution under relatively mild conditions. It is a generalization of Peano's existence theorem...
, about the existence of solutions to ordinary differential equations - Carathéodory's extension theoremCarathéodory's extension theoremIn measure theory, Carathéodory's extension theorem states that any σ-finite measure defined on a given ring R of subsets of a given set Ω can be uniquely extended to the σ-algebra generated by R...
, about the extension of a measure - Borel–Carathéodory theoremBorel–Carathéodory theoremIn mathematics, the Borel–Carathéodory theorem in complex analysis shows that an analytic function may be bounded by its real part. It is an application of the maximum modulus principle. It is named for Émile Borel and Constantin Carathéodory....
, about the boundedness of a complex analytic function - Carathéodory–Jacobi–Lie theorem, a generalization of Darboux's theorem in symplectic topology