Carathéodory conjecture

In differential geometry, the Carathéodory conjecture is a mathematical conjecture attributed to Constantin Carathéodory by Hans Ludwig Hamburger in a session of the Berlin Mathematical Society in 1924. Carathéodory did publish a paper on a related subject, but never committed the Conjecture into writing. In,John Edensor Littlewood mentions the Conjecture and Hamburger's contribution as an example of a mathematical claim that is easy to state but difficult to prove. Dirk Struik describes in the formal analogy of the Conjecture with the Four Vertex Theorem for plane curves. Modern references to the Conjecture are the problem list of Shing-Tung Yau, the books of Marcel Berger, as well as the books.

The Conjecture claims that any convex, closed and sufficiently smooth surface in three dimensional Euclidean space needs to admit at least two umbilic points. In the sense of the Conjecture, the spheroid with only two umbilic points and the sphere, all points of which are umbilic, are examples of surfaces with minimal and maximal numbers of umbilics. For the conjecture to be well posed, or the umbilic points to be well-defined, the surface needs to be at least twice differentiable.

The invited address of Stefan Cohn-Vossen to the International Congress of Mathematicians of 1928 in Bologna was on the subject and in the 1929 edition of Wilhelm Blaschke's third volume on Differential Geometry he states:

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