In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Carathéodory's theorem may refer to one of a number of results of
Constantin Carathéodory
Constantin Carathéodory (; 13 September 1873 – 2 February 1950) was a Greeks, Greek mathematician who spent most of his professional career in Germany. He made significant contributions to real and complex analysis, the calculus of variations, ...
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Carathéodory's theorem (conformal mapping) In mathematics, Carathéodory's theorem is a theorem in complex analysis, named after Constantin Carathéodory, which extends the Riemann mapping theorem. The theorem, published by Carathéodory in 1913, states that any conformal mapping sending the ...
, about the extension of conformal mappings to the boundary
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Carathéodory's theorem (convex hull)
Carathéodory's theorem is a theorem in convex geometry. It states that if a point x lies in the convex hull \mathrm(P) of a set P\subset \R^d, then x lies in some ''d''-dimensional simplex with vertices in P. Equivalently, x can be written as the ...
, about the convex hulls of sets in
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Carathéodory's existence theorem
In 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. Peano's theorem requires that the right-hand sid ...
, about the existence of solutions to ordinary differential equations
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Carathéodory's extension theorem, about the extension of a measure
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Carathéodory–Jacobi–Lie theorem, a generalization of Darboux's theorem in symplectic topology
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Carathéodory's criterion
Carathéodory's criterion is a result in measure theory that was formulated by Greek mathematician Constantin Carathéodory that characterizes when a set is Lebesgue measurable.
Statement
Carathéodory's criterion:
Let \lambda^* : (\R^n) \to ...
, a necessary and sufficient condition for a
measurable set
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts hav ...
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Carathéodory kernel theorem, a geometric criterion for local uniform convergence of univalent functions
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Borel–Carathéodory theorem In 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
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Vitali–Carathéodory theorem, a result in real analysis
Mathematics disambiguation pages
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