The Carathéodory family, originally from Bosnochori or Vyssa, was well established and respected in Constantinople, and its members held many important governmental positions.

The Carathéodory family spent 1874–75 in Constantinople, where Constantin's paternal grandfather lived, while his father Stephanos was on leave.

This put Carathéodory in a difficult position since he sided with the Greeks, yet his father served the government of the Ottoman Empire.

During periods when construction work had to stop due to floods, he studied mathematics from some textbooks he had with him, such as Jordan's Cours d'Analyse and Salmon's text on the analytic geometry of conic sections.

[6] Carathéodory studied engineering in Belgium at the Royal Military Academy, where he was considered a charismatic and brilliant student.

During the difficult period of World War II, his close associates at the Bavarian Academy of Sciences were Perron and Tietze.

Einstein, then a member of the Prussian Academy of Sciences in Berlin, was working on his general theory of relativity when he contacted Carathéodory for clarifications on the Hamilton-Jacobi equation and canonical transformations.

Einstein told Carathéodory his derivation was "beautiful" and recommended its publication in the Annalen der Physik.

[8] While in Germany, Carathéodory retained numerous links with the Greek academic world, details of which can be found in Georgiadou's book.

Kritikos and Carathéodory helped the Greek topologist Christos Papakyriakopoulos take a doctorate in topology at Athens University in 1943 under very difficult circumstances.

While teaching at Athens University, Carathéodory had Evangelos Stamatis as an undergraduate student, who subsequently achieved considerable distinction as a scholar of ancient Greek mathematical classics.

[9] In his doctoral dissertation, Carathéodory showed how to extend solutions to discontinuous cases and studied isoperimetric problems.

[8] Previously, between the mid-1700s to the mid-1800s, Leonhard Euler, Adrien-Marie Legendre, and Carl Gustav Jacob Jacobi were able to establish necessary but insufficient conditions for the existence of a strong relative minimum.

[10][11] A key advantage of Carathéodory's work on this topic is that it illuminates the relation between the calculus of variations and partial differential equations.

He published his Variationsrechnung und Partielle Differentialgleichungen Erster Ordnung (Calculus of Variations and First-order Partial Differential Equations) in 1935.

[10] More recently, Carathéodory's work on the calculus of variations and the Hamilton-Jacobi equation has been taken into the theory of optimal control and dynamic programming.

[13] He is credited with the authorship of the Carathéodory conjecture claiming that a closed convex surface admits at least two umbilic points.

[14] [15] [16] He proved an existence theorem for the solution to ordinary differential equations under mild regularity conditions.

"[22] Max Born acclaimed this "first axiomatically rigid foundation of thermodynamics" and he expressed his enthusiasm in his letters to Einstein.

[23][19] However, Max Planck had some misgivings[24] in that while he was impressed by Carathéodory's mathematical prowess, he did not accept that this was a fundamental formulation, given the statistical nature of the second law.

He argued that an important advantage of his approach was that it covers the integral invariants of Henri Poincaré and Élie Cartan and completes the Malus law.

He explained that in his investigations in optics, Pierre de Fermat conceived a minimum principle similar to that enunciated by Hero of Alexandria to study reflection.

[28] During the Second World War Carathéodory edited two volumes of Euler's Complete Works dealing with the Calculus of Variations which were submitted for publication in 1946.

In 1920 Carathéodory was appointed dean of the university and took a major part in establishing the institution, touring Europe to buy books and equipment.

The university however never actually admitted students, due to the War in Asia Minor which ended in the Great Fire of Smyrna.

Carathéodory managed to save books from the library and was only rescued at the last moment by a journalist who took him by rowboat to the battleship Naxos which was standing by.

Such an impressive linguistic arsenal enabled him to communicate and exchange ideas directly with other mathematicians during his numerous travels, and greatly extended his fields of knowledge.

[35] In the town of Nea Vyssa, Caratheodory's ancestral home, a unique family museum is to be found.

The museum is located in the central square of the town near to its church, and includes a number of Karatheodory's personal items, as well as letters he exchanged with Albert Einstein.

[39][40][41] A complete list of Carathéodory's journal article publications can be found in his Collected Works(Ges.