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. He attended the Ecole Militaire de Belgique from October 1891 to May 1895 and he also studied at the Ecole d’Application from 1893 to 1896. In 1897 a war broke out between Turkey and Greece. This put Caratheodory in a difficult position since he sided with the Greeks, yet his father served the government of the Ottoman Empire.

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

Caratheodory is credited with the theories of outer measure, and prime ends, amongst other mathematical results. He is credited with the authorship of the Caratheodory conjecture claiming that a closed convex surface admits at least two umbilic points. In 1909, Caratheodory published a pioneering work “Investigations on the Foundations of Thermodynamics” in which he formulated the Laws of Thermodynamics axiomatically, using only mechanical concepts and the theory of Pfaff’s differential forms. He expressed the Second Law of Thermodynamics via the following Axiom: “In the neighbourhood of any initial state, there are states which cannot be approached arbitrarily close through adiabatic changes of state.” Caratheodory coined the term adiabatic accessibility. This “first axiomatically rigid foundation of thermodynamics” was acclaimed by Max Planck and Max Born.

On 20 October 1919, he submitted a plan for the creation of a new University in Greece, to be named Ionian University. This university never actually admitted students due to the War in Asia Minor in 1922, but the present day University of the Aegean claims to be a continuation of Caratheodory’s original plan.

In 1920 Caratheodory accepted a post in the University of Smyrna, invited by Prime Minister Eleftherios Venizelos. He took a major part in establishing the institution, but his efforts 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 university library, thus saving it from destruction. He stayed at Athens and taught at the university and technical school until 1924.

In 1924 Caratheodory was appointed professor of mathematics at the University of Munich, and held this position 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

In 2005, Israeli officials presented the Greek foreign ministry with copies of 10 letters between Albert Einstein and Constantin Caratheodory that suggest that the work of Caratheodory helped shape some of Albert Einstein’s theories. The letters were part of a long correspondence which lasted from 1916 to 1930. According to experts at the National Archives of Israel – custodians of the original letters – the mathematical side of Einstein’s physics theory was partly substantiated through the work of Caratheodory.