In mathematics, Carathéodory's theorem may refer to one of a number of results of Constantin Carathéodory: Carathéodory's theorem (conformal mapping), about the extension of conformal mappings to the boundary Carathéodory's theorem (convex hull), about the convex hulls of sets in Euclidean space Carathéodory's existence theorem, about the existence of solutions to ordinary differential equations Carathéodory's extension theorem, about the extension of a measure Carathéodory–Jacobi–Lie theorem, a generalization of Darboux's theorem in symplectic topology Carathéodory's criterion, a necessary and sufficient condition for a measurable set Carathéodory kernel theorem, a geometric criterion for local uniform convergence of univalent functions Borel–Carathéodory theorem, about the boundedness of a complex analytic function Vitali–Carathéodory theorem, a result in real analysis Mathematics disambiguation pages