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Constantin Caratheodory (or Constantine Karatheodoris), born September 13, 1873, in Berlin, died February 2, 1950, was a mathematician who made significant contributions to several mathematical theories such as the theory of functions of a real variable, the calculus of variations, and measure theory.

In 1909, Caratheodory pioneered the Axiomatic Formulation of Thermodynamics along with a purely geometrical approach.

Caratheodory’s parents were Greeks from Constantinople. His father Stephanos served as the Ottoman ambassador to Belgium (Brussels), where Caratheodory grew up (he was born in Berlin).

In 1895 Caratheodory lost his beloved mother Despina who died of pneumonia in Cannes. Constantin’s maternal grandmother took on the task of raising Constantin and his sister Loulia in his father’s home in Belgium.

After high school Caratheodory began training as a military engineer. Caratheodory  attended the Ecole Militaire de Belgique (from 1891 to 1895) while at the same time he also studied at the Ecole d’Application (until 1896). When the war between Turkey and Greece broke in 1897 Caratheodory sided with the Greeks but this put him (and his family) in a difficult position since at the time his father served the government of the Ottoman Empire.

Caratheodory studied engineering in Belgium, where he was considered a charismatic and brilliant student, and in 1900 he entered the University of Berlin. and then eventually completed his graduate studies in the University of Gottingen in 1902. Caratheodory held various lecturing positions in Bonn, Hannover, Breslau, Gottingen, and Berlin from 1908-1920.

Caratheodory is credited with the theories of outer measure, and prime ends, amongst other mathematical results and also with a conjecture claiming that a closed convex surface admits at least two umbilic points. He then published a pioneering work in 1909 by the title “Investigations on the Foundations of Thermodynamics”.

On 20 October 1919, he submitted a plan for the creation of a new University in Greece, to be named Ionian University. Due to the war in Asia Minor between Greece and Turkey in 1922 this university never admitted students , but it’s claimed that the present day University of the Aegean is a continuation of Caratheodory’s original plan.

When Smyrna was liberated by the Greeks (1920) Prime Minister of Greece Eleftherios Venizelos invited Caratheodory to take a post in the University of Smyrna, which Caratheodory proudly accepted. He worked hard in establishing the institution, but his efforts ufortunately ended in 1922 when the Greek population was expelled from the city during the War in Asia Minor.

Having been forced to move to Athens, Caratheodory brought along with him some of the Smyrna university library, thus saving it from destruction. Caratheodory stayed in Athens and taught at the university of Athens and the technical school until 1924.

Caratheodory was then appointed as a professor of mathematics at the University of Munich in 1924, a position which he held until his death in 1950.

Caratheodory formulated the axiomatic principle of irreversibility in thermodynamics in 1909, stating that inaccessibility of states is related to the existence of entropy, where temperature is the integration function.

In 1926 he gave a strict and general proof, that no system of lenses and mirrors can avoid aberration, except for the trivial case of plane mirrors.

In convex geometry, Carathéodory's theorem states that if a point x of Rd lies in the convex hull of a set P, then x can be written as the convex combination of at most d + 1 points in P. Namely, there is a subset P′ of Pconsisting 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 {displaystyle rleq d}. The smallest r that makes the last statement valid for each x in the convex hull of P is defined as the Carathéodory's number of P. Depending on the properties of P, upper bounds lower than the one provided by Carathéodory's theorem can be obtained.Note that P need not be itself convex. A consequence of this is that P′ can always be extremal in P, as non-extremal points can be removed from P without changing the membership of x in the convex hull.

Carathéodory's theorem

Greek and French were Caratheodory’s first languages, and he mastered German with such perfection, that his writings composed in the German language are stylistic masterworks. Caratheodory also spoke and wrote English, Italian, and Turkish. Such an impressive linguistic arsenal enabled him to communicate and exchange ideas directly with other mathematicians during his numerous travels, and greatly extend his fields of knowledge.

Much more than that, Caratheodory was a treasured conversation partner for his fellow professors in the Munich Department of Philosophy. The well-respected, German philologist Kurt von Fritz praised Caratheodory, saying that from him one could learn an endless amount about the old and new Greece, the old Greek language, and Hellenic mathematics. Caratheodory loved his Greek heritage, and deep in his heart he felt Greek above all. Greek  was the only language spoken in the Caratheodory’s house by himslef, and his children, his son Stephanos and daughter Despina.

Caratheodory and Einstein

Caratheodory and Einstein corresponded frquently between 1916 and 1930. Several of these letters ended up at the National Archives of Israel and in 2005 Israel presented the Greek foreign ministry with copies of 10 letters between Albert Einstein and Constantin Caratheodory which suggest that the work of Caratheodory helped shape some of Albert Einstein’s ideas and theories. According to experts at the National Archives of Israel the mathematical side of Einstein’s physics theory was partly substantiated through the work of Caratheodory.

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