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

His work also includes important results in conformal representations and in the theory of boundary correspondence. In 1909, Caratheodory pioneered the Axiomatic Formulation of Thermodynamics along with a purely geometrical approach.


Constantin Caratheodory was born in Berlin to Greek parents and grew up in Brussels, where his father served as the Ottoman ambassador to Belgium. The Caratheodory family was well-established and respected in Constantinople, and its members held many important governmental positions.

The Caratheodory family spent 1874-75 in Constantinople, where Constantin’s paternal grandfather lived, while Constantine’s father Stephanos was on leave. Then in 1875, they went to Brussels when Stephanos was appointed there as Ottoman Ambassador. In Brussels, Constantin’s younger sister Loulia was born. The year 1895 was a tragic one for the family since Constantin’s paternal grandfather died in that year, but much more tragically, Constantin’s mother Despina died of pneumonia in Cannes. Constantin’s maternal grandmother took on the task of bringing up Constantin and Loulia in his father’s home in Belgium. They employed a German maid who taught the children to speak German. Constantin was already bilingual in French and Greek by this time.

Constantin began his formal schooling at a private school in Vanderstock in 1881. He left after two years and then spent time with his father on a visit to Berlin, and also spent the winters of 1883-84 and 1884-85 on the Italian Riviera. Back in Brussels in 1885, he attended a grammar school for a year where he first began to become interested in mathematics. In 1886 he entered the high school Athenee Royal d’Ixelles and studied there until his graduation in 1891. Twice during his time at this school, Constantin won a prize as the best mathematics student in Belgium.

At this stage, 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. Since he was a trained engineer he was offered a job in the British colonial service. This job took him to Egypt where he worked on the construction of the Assiut dam until April 1900. 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. He also visited the Cheops pyramid and made measurements which he wrote up and published in 1901. He also published a book on Egypt in the same year which contained a wealth of information on the history and geography of the country.


Caratheodory studied engineering in Belgium, where he was considered a charismatic and brilliant student. In 1900 he entered the University of Berlin. In the years 1902-1904, he completed his graduate studies in the University of Gottingen under the supervision of Hermann Minkowski. During the years 1908-1920, he held various lecturing positions in Bonn, Hannover, Breslau, Gottingen, and Berlin.


He 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. As of 2007, this conjecture remained unproven despite having attracted a large amount of research. In 1909, Caratheodory published a pioneering work “Investigations on the Foundations of Thermodynamics” (Untersuchungen ueber die Grundlagen der Thermodynamik, Math. Ann., 67 (1909) p. 355-386) 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.

The Smyrna Years

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

Linguistic Talent

Caratheodory excelled at languages, much like many members of his family did. Greek and French were his 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, Turkish, and the ancient languages without any effort. 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, professor of ancient languages 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. Fritz had an uncountable number of philosophical discussions with Caratheodory. Deep in his heart, Caratheodory felt Greek above all. The Greek language was spoken exclusively in Caratheodory’s house – his son Stephanos and daughter Despina went to a German high school, but they obtained daily additional instruction in Greek language and culture from a Greek priest. At home, they were not allowed to speak any other language.

The Einstein Connection

On December 19, 2005, Israeli officials along with Israel’s ambassador to Athens, Ram Aviram, presented the Greek foreign ministry with copies of 10 letters between Albert Einstein and Constantin Caratheodory [Karatheodoris] 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. Aviram said that 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.