Constantin Carathéodory

For the Ottoman Greek doctor, see Constantine Caratheodory (1802-1879).

Constantin Carathéodory (Greek: Κωνσταντίνος Καραθεοδωρή, romanized: Konstantinos Karatheodori; 13 September 1873 – 2 February 1950) was a Greek mathematician who spent most of his professional career in Germany. He made significant contributions to real and complex analysis, the calculus of variations, and measure theory. He also created an axiomatic formulation of thermodynamics. Carathéodory is considered one of the greatest mathematicians of his era^[3] and the most renowned Greek mathematician since antiquity.^[4]

Constantin Carathéodory
Constantin Carathéodory
Born	(1873-09-13)13 September 1873 Berlin, German Empire
Died	2 February 1950(1950-02-02) (aged 76) Munich, West Germany
Nationality	Greek
Alma mater	University of Berlin University of Göttingen
Known for	Carathéodory conjecture Carathéodory function Carathéodory metric Carathéodory theorems Carathéodory's criterion Carathéodory's lemma Carathéodory's positivity criterion for holomorphic functions Carathéodory's principle Carnot–Carathéodory metric Adiabatic accessibility Cyclic polytope Prime end General theory of outer measures Axiomatic formulation of thermodynamics
Scientific career
Fields	Calculus of variations Real analysis Complex analysis Measure theory
Institutions	University of Bonn Hannover Technical High School Breslau Technical High School University of Göttingen University of Berlin University of Munich National Technical University of Athens Ionian University of Smyrna
Doctoral advisor	Hermann Minkowski^[1]
Doctoral students	Paul Finsler Hans Rademacher Georg Aumann Hermann Boerner Ernst Peschl Wladimir Seidel Nazım Terzioğlu^[2]
Signature

Origins

 

Carathéodory with his father, Stephanos, in 1900.

 

Carathéodory (left) pictured sitting with his father, brother in law and sister, Carlsbad 1898

Constantin Carathéodory was born in 1873 in Berlin to Greek parents and grew up in Brussels. His father Stephanos [tr], a lawyer, served as the Ottoman ambassador to Belgium, St. Petersburg and Berlin. His mother, Despina, née Petrokokkinos, was from the island of Chios. 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. Then in 1875 they went to Brussels when Stephanos was appointed there as Ottoman Ambassador. In Brussels, Constantin's younger sister Julia was born. The year 1879 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 Julia 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 Athénée 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 Carathéodory began training as a military engineer. He attended the École Militaire de Belgique from October 1891 to May 1895 and he also studied at the École d'Application from 1893 to 1896. In 1897 a war broke out between the Ottoman Empire and Greece. This put Carathéodory 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.^[5] 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.^[6]

Studies and university career

 

Young Carathéodory

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

University career

• 1900 Studies at University of Berlin.
• 1902 Completed graduation at University of Göttingen (1904 Ph.D, 1905 Habilitation)
• 1908 Dozent at Bonn
• 1909 Ordinary Professor at Hannover Technical High School.
• 1910 Ordinary Professor at Breslau Technical High School.
• 1913 Professor following Klein at University of Göttingen.
• 1919 Professor at University of Berlin
• 1919 Elected to Prussian Academy of Science.
• 1920 University Dean at Ionian University of Smyrna (later, University of the Aegean).
• 1922 Professor at University of Athens.
• 1922 Professor at Athens Polytechnic.
• 1924 Professor following Lindemann at University of Munich.
• 1938 Retirement from Professorship. Continued working from Bavarian Academy of Science

Doctoral students

Carathéodory had about 20 doctoral students among these being Hans Rademacher, known for his work on analysis and number theory, and Paul Finsler known for his creation of Finsler space.

Academic contacts in Germany

 

Carathéodory (left) with Hungarian mathematician Lipót Fejér (1880–1959) (standing to the right).

Carathéodory's contacts in Germany were many and included such famous names as: Hermann Minkowski, David Hilbert, Felix Klein, Albert Einstein, Edmund Landau, Hermann Amandus Schwarz, Lipót Fejér. During the difficult period of World War II his close associates at the Bavarian Academy of Sciences were Perron and Tietze.

Einstein, then in a member of the Prussian Academy of Sciences in Berlin, was working on his general theory of relativity when he contacted Carathéodory asking for clarifications on the Hamilton-Jacobi equation and canonical transformations. He wanted to see a satisfactory derivation of the former and the origins of the latter. Einstein told Carathéodory his derivation was "beautiful" and recommended its publication in the Annalen der Physik. Einstein employed the former in a 1917 paper titled Zum Quantensatz von Sommerfeld und Epstein (On the Quantum Theorem of Sommerfeld and Epstein). Carathéodory explained some fundamental details of the canonical transformations and referred Einstein to E.T. Whittaker's Analytical Dynamics. Einstein was trying to solve the problem of "closed time-lines" or the geodesics corresponding to the closed trajectory of light and free particles in a static universe, which he introduced in 1917.^[7]

Landau and Schwarz stimulated his interest in the study of complex analysis.^[8]

Academic contacts in Greece

While in Germany Carathéodory retained numerous links with the Greek academic world about which detailed information may be found in Georgiadou's book. He was directly involved with the reorganization of Greek universities. An especially close friend and colleague in Athens was Nicolaos Kritikos who had attended his lectures at Göttingen, later going with him to Smyrna, then becoming professor at Athens Polytechnic. 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 in Athens University Carathéodory had as undergraduate student Evangelos Stamatis who subsequently achieved considerable distinction as a scholar of ancient Greek mathematical classics.^[9]

Works

Calculus of variations

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. In 1879, Karl Weierstrass added a fourth that does indeed guarantee such a quantity exists. ^[10] Carathéodory constructed his method for deriving sufficient conditions based on the use of the Hamilton–Jacobi equation to construct a field of extremals. The ideas are closely related to light propagation in optics. The method became known as Carathéodory's method of equivalent variational problems or the royal road to the calculus of variations.^[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.^[8] It allows for quick and elegant derivations of conditions of sufficiency in the calculus of variations and leads directly to the Euler-Lagrange equation and the Weierstrass condition. 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.^[10][12] The method can also be extended to multiple integrals.^{[citation needed]}

Convex geometry

 

An illustration of Carathéodory's theorem (convex hull) for a square in R².

Carathéodory's theorem in convex geometry states that if a point ${\displaystyle x}$  of ${\displaystyle \mathbb {R} ^{d}}$  lies in the convex hull of a set ${\displaystyle P}$ , then ${\displaystyle x}$  can be written as the convex combination of at most ${\displaystyle d+1}$  points in ${\displaystyle P}$ . Namely, there is a subset ${\displaystyle P'}$  of ${\displaystyle P}$  consisting of ${\displaystyle d+1}$  or fewer points such that ${\displaystyle x}$  lies in the convex hull of ${\displaystyle P'}$ . Equivalently, ${\displaystyle x}$  lies in an ${\displaystyle r}$ -simplex with vertices in ${\displaystyle P}$ , where ${\displaystyle r\leq d}$ . The smallest ${\displaystyle r}$  that makes the last statement valid for each ${\displaystyle x}$  in the convex hull of P is defined as the Carathéodory's number of ${\displaystyle P}$ . Depending on the properties of ${\displaystyle P}$ , upper bounds lower than the one provided by Carathéodory's theorem can be obtained.^[13]

He is credited with the authorship of the Carathéodory conjecture claiming that a closed convex surface admits at least two umbilic points. As of 2021, this conjecture remained unproven despite having attracted a large amount of research.

Real analysis

He proved an existence theorem for the solution to ordinary differential equations under mild regularity conditions.

Another theorem of his on the derivative of a function at a point could be used to prove the Chain Rule and the formula for the derivative of inverse functions.^[14]

Complex analysis

He greatly extended the theory of conformal transformation^[15] proving his theorem about the extension of conformal mapping to the boundary of Jordan domains. In studying boundary correspondence he originated the theory of prime ends.^[8] He exhibited an elementary proof of the Schwarz lemma.^[8]

Carathéodory was also interested in the theory of functions of multiple complex variables. In his investigations on this subject he sought analogs of classical results from the single-variable case. He proved that a ball in ${\displaystyle \mathbb {C} ^{2}}$  is not holomorphically equivalent to the bidisc.^[8]

Theory of measure

He is credited with the Carathéodory extension theorem which is fundamental to modern measure theory. Later Carathéodory extended the theory from sets to Boolean algebras.

Thermodynamics

Thermodynamics had been a subject dear to Carathéodory since his time in Belgium.^[16] In 1909, he published a pioneering work "Investigations on the Foundations of Thermodynamics"^[17] in which he formulated the second law of thermodynamics axiomatically, that is, without the use of Carnot engines and refrigerators and only by mathematical reasoning. This is yet another version of the second law, alongside the statements of Clausius, and of Kelvin and Planck.^[18] Carathéodory's version attracted the attention of some of the top physicists of the time, including Max Planck, Max Born, and Arnold Sommerfeld.^[8] According to Bailyn's survey of thermodynamics, Carathéodory's approach is called "mechanical," rather than "thermodynamic."^[19] Max Born acclaimed this "first axiomatically rigid foundation of thermodynamics" and he expressed his enthusiasm in his letters to Einstein.^[20][16] However, Max Planck had some misgivings^[21] 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.^[16]

In his theory he simplified the basic concepts, for instance heat is not an essential concept but a derived one.^[22] He formulated the axiomatic principle of irreversibility in thermodynamics stating that inaccessibility of states is related to the existence of entropy, where temperature is the integration function. The Second Law of Thermodynamics was expressed 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." In this connexion he coined the term adiabatic accessibility.^[23]

Optics

Carathéodory's work in optics is closely related to his method in the calculus of variations. 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 his later work he gave the theory of the Schmidt telescope.^[24] In his Geometrische Optik (1937), Carathéodory demonstrated the equivalence of Huygens' principle and Fermat's principle starting from the former using Cauchy's theory of characteristics. 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.^[25]

Historical

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.^[26]

The University of Smyrna

 

Photo of the Ionian University of Smyrna.

At the time, Athens was the only major educational centre in the wider area and had limited capacity to sufficiently satisfy the growing educational need of the eastern part of the Aegean Sea and the Balkans. Constantin Carathéodory, who was a professor at the University of Berlin at the time, proposed the establishment of a new University ^[27] - the difficulties regarding the establishment of a Greek university in Constantinople led him to consider three other cities: Thessaloniki, Chios and Smyrna.^[28]

At the invitation of the Greek Prime Minister Eleftherios Venizelos he submitted a plan on 20 October 1919 for the creation of a new University at Smyrna in Asia Minor, to be named Ionian University of Smyrna. 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.^[29] Carathéodory brought to Athens some of the university library and stayed in Athens, teaching at the university and technical school until 1924.

In 1924 Carathéodory was appointed professor of mathematics at the University of Munich, and held this position until retirement in 1938. He later worked from the Bavarian Academy of Sciences until his death in 1950.

The new Greek University in the broader area of the Southeast Mediterranean region, as originally envisioned by Carathéodory, finally materialised with the establishment of the Aristotle University of Thessaloniki in 1925.^[30]

Linguistic and oratorical talents

 

Caratheodory at a mature age.

Carathéodory excelled at languages, much like many members of his family. 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.^[31] Carathéodory 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 extended his fields of knowledge.

Much more than that, Carathéodory was a treasured conversation partner for his fellow professors in the Munich Department of Philosophy. The well-respected German philologist and professor of ancient languages, Kurt von Fritz, praised Carathéodory on the grounds that from him one could learn an endless amount about the old and new Greece, the old Greek language, and Hellenic mathematics. Von Fritz conducted numerous philosophical discussions with Carathéodory.

The mathematician sent his son Stephanos and daughter Despina to a German high school, but they also obtained daily additional instruction in Greek language and culture from a Greek priest, and at home he allowed them to speak Greek only.

Carathéodory was a talented public speaker, and was often invited to give speeches. In 1936, it was he who handed out the first ever Fields Medals at the meeting of the International Congress of Mathematicians in Oslo, Norway.^[8]

Legacy

 

Grave of Carathéodory in Munich.

In 2002, in recognition of his achievements, the University of Munich named one of the largest lecture rooms in the mathematical institute the Constantin-Carathéodory Lecture Hall.^[32]

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. More information is provided at the original website of the club, http://www.s-karatheodoris.gr.

At the same time, Greek authorities had long since intended to create a museum honoring Karatheodoris in Komotini, a major town of the northeastern Greek region, more than 200km away from his home town above. On 21 March 2009, the "Karatheodoris" Museum (Καραθεοδωρής) opened its gates to the public in Komotini.^[33][34][35]

The coordinator of the Museum, Athanasios Lipordezis (Αθανάσιος Λιπορδέζης), has noted that the museum provides a home for original manuscripts of the mathematician running to about 10,000 pages, including correspondence with the German mathematician Arthur Rosenthal for the algebraization of measure. At the showcase, visitors are also able to view the books " Gesammelte mathematische Schriften Band 1,2,3,4 ", "Mass und ihre Algebraiserung", " Reelle Functionen Band 1", " Zahlen/Punktionen Funktionen ", and a number of others. Handwritten letters by Carathéodory to Albert Einstein and Hellmuth Kneser, as well as photographs of the Carathéodory family, are on display.

Efforts to furnish the museum with more exhibits are ongoing.^[36][37][38]

Publications

Journal articles

A complete list of Carathéodory's journal article publications can be found in his Collected Works(Ges. Math. Schr.). Notable publications are:

• Über die kanonischen Veränderlichen in der Variationsrechnung der mehrfachen Integrale^[39]
• Über das Schwarzsche Lemma bei analytischen Funktionen von zwei komplexen Veränderlichen^[40]
• Über die diskontinuierlichen Lösungen in der Variationsrechnung. Diss. Göttingen Univ. 1904; Ges. Math. Schr. I 3–79.
• Über die starken Maxima und Minima bei einfachen Integralen. Habilitationsschrift Göttingen 1905; Math. Annalen 62 1906 449–503; Ges. Math. Schr. I 80–142.^[41]
• Untersuchungen über die Grundlagen der Thermodynamik, Math. Ann. 67 (1909) pp. 355–386; Ges. Math. Schr. II 131–166.^[42]
• Über das lineare Mass von Punktmengen – eine Verallgemeinerung des Längenbegriffs., Gött. Nachr. (1914) 404–406; Ges. Math. Schr. IV 249–275.
• Elementarer Beweis für den Fundamentalsatz der konformen Abbildungen. Schwarzsche Festschrift, Berlin 1914; Ges. Math. Schr.IV 249–275.^[43]
• Zur Axiomatic der speziellen Relativitätstheorie. Sitzb. Preuss. Akad. Wiss. (1924) 12–27; Ges. Math. Schr. II 353–373.
• Variationsrechnung in Frank P. & von Mises (eds): Die Differential= und Integralgleichungen der Mechanik und Physik, Braunschweig 1930 (Vieweg); New York 1961 (Dover) 227–279; Ges. Math. Schr. I 312–370.
• Entwurf für eine Algebraisierung des Integralbegriffs, Sitzber. Bayer. Akad. Wiss. (1938) 27–69; Ges. Math. Schr. IV 302–342.

Books

• Carathéodory, Constantin (1918), Vorlesungen über reelle Funktionen (3rd ed.), Leipzig: Teubner, ISBN 978-0-8284-0038-1, MR 0225940 Reprinted 1968 (Chelsea)
• Conformal Representation, Cambridge 1932 (Cambridge Tracts in Mathematics and Physics)
• Geometrische Optik, Berlin, 1937
• Elementare Theorie des Spiegelteleskops von B. Schmidt (Elementary Theory of B. Schmidt's Reflecting Telescope), Leipzig Teubner, 1940 36 pp.; Ges. math. Schr. II 234–279
• Funktionentheorie I, II, Basel 1950,^[44] 1961 (Birkhäuser). English translation: Theory of Functions of a Complex Variable, 2 vols, New York, Chelsea Publishing Company, 3rd ed 1958
• Mass und Integral und ihre Algebraisierung, Basel 1956. English translation, Measure and Integral and Their Algebraisation, New York, Chelsea Publishing Company, 1963
• Variationsrechnung und partielle Differentialgleichungen erster Ordnung, Leipzig, 1935. English translation next reference
• Calculus of Variations and Partial Differential Equations of the First Order, 2 vols. vol. I 1965, vol. II 1967 Holden-Day.
• Gesammelte mathematische Schriften München 1954–7 (Beck) I–V.

See also

•  Biography portal
•  Mathematics portal
•  Physics portal

• Domain (mathematical analysis)
• Nemytskii operator
• Herbert Callen, who also sought an axiomatic formulation of thermodynamics

Notes

1. . Mathematics Genealogy Project. North Dakota State University Department of Mathematics. Archived from the original on 13 July 2018. Retrieved 27 August 2017.
2. "The Mathematics Genealogy Project - Nazım Terzioğlu". Mathematics Genealogy Project. North Dakota State University Department of Mathematics. Retrieved 27 August 2017.
3. Hallett, Michael; Majer, Ulrich (2004). David Hilbert's Lectures on the Foundations of Geometry 1891–1902. Springer Science & Business Media. p. 11. ISBN 978-3-540-64373-9.
4. Szpiro, George G. (2008). Poincare's Prize: The Hundred-Year Quest to Solve One of Math's Greatest Puzzles. Penguin. p. 104. ISBN 978-1-4406-3428-4.
5. Brussells 1901 (Hayez);Ges. math. Schr. V. 273-281
6. H Aigyptos, Syllogos Ophelimon Biblion, no 14, 118 pp Athens 1901, 1928, New York 1920
7. Georgiadou, Maria (2004). "2.15: Einstein Contacts Carathéodory". Constantin Carathéodory: Mathematics and Politics in Turbulent Times. Germany: Springer. ISBN 3-540-20352-4.
8. Begehr, H. G. W. (1998). "Constantin Carathéodory (1873-1950)". In Begehr, H. G. W.; Koch, H; Krammer, J; Schappacher, N; Thiele, E.-J (eds.). Mathematics in Berlin. Germany: Birkhäuser Verlag. ISBN 3-7643-5943-9.
9. J P Christianidis & N Kastanis: In memoriam Evangelos S Stamatis (1898–1990) Historia Mathematica 19 (1992) 99-105
10. Kot, Mark (2014). "Chapter 12: Sufficient Conditions". A First Course in the Calculus of Variations. American Mathematical Society. ISBN 978-1-4704-1495-5.
11. H. Boerner, Carathéodory und die Variationsrechnung, in A Panayotopolos (ed.), Proceedings of C. Carathéodory International Symposium, September 1973, Athens (Athens, 1974), 80–90.
12. Bellman for his Dynamic programming in its continuous-time form used Carathéodory's work in the form of the Hamilton–Jacobi–Bellman equation. Kálmán also explicitly used Carathéodory's formulation in his initial papers on optimal control. See e.g. R. E. Kalman: Contributions to the theory of optimal control. Boletin de la Sociedad Matematica Mexicana 1960
13. Bárány, Imre; Karasev, Roman (2012-07-20). "Notes About the Carathéodory Number". Discrete & Computational Geometry. 48 (3): 783–792. arXiv:1112.5942. doi:10.1007/s00454-012-9439-z. ISSN 0179-5376. S2CID 9090617.
14. Bartle, Robert G.; Sherbert, Donald R. (2011). "6.1: The Derivative". Introduction to Real Analysis. John Wiley & Sons. ISBN 978-0-471-43331-6.
15. A. Shields: Carathéodory and Conformal Mapping Math. Intelligencer vol.10(1), 1988
16. Georgiadou, Maria (2004). "2.2 Axiomatic Foundation of Thermodynamics". Constantin Carathéodory: Mathematics and Politics in Turbulent Times. Germany: Springer. ISBN 3-540-20352-4.
17. Carathéodory, Constantin (1909). Translated by Delphinich, D. H. [Examination of the foundations of Thermodynamics] (PDF). Mathematische Annalen. 67 (3): 355–386. doi:10.1007/bf01450409. S2CID 118230148. Archived from the original (PDF) on 2019-10-12. Retrieved 2016-07-09.
18. Lewis, Christopher J. T. (2007). "Chapter 5. Energy and Entropy: The Birth of Thermodynamics.". Heat and Thermodynamics: A Historical Perspective. Westport, Connecticut: Greenwood Press. p. 110. ISBN 978-0-313-33332-3.
19. Bailyn, M. (1994). A Survey of Thermodynamics, American Institute of Physics, Woodbury NY, ISBN 0-88318-797-3.
20. Max Born: The Born–Einstein Letters, MacMillan 1971
21. Constantin Carathéodory and the axiomatic thermodynamics by Lionello Pogliani and Mario N. Berberan-Santos
22. Pogliani, Lionello; Berberan-Santos, Mario N. (2000). "Constantin Carathéodory and the axiomatic thermodynamics". Journal of Mathematical Chemistry. 28 (1/3): 313–324. doi:10.1023/A:1018834326958. S2CID 17244147.
23. adiabatic accessibility = adiabatische Erreichbarkeit; see also Elliott H. Lieb, Jakob Yngvason: The Physics and Mathematics of the Second Law of Thermodynamics, Phys. Rep. 310, 1–96 (1999) and Elliott H. Lieb, (editors: B. Nachtergaele, J.P. Solovej, J. Yngvason): Statistical Mechanics: Selecta of Elliott H. Lieb, 2005, ISBN 978-3-540-22297-2
24. Über den Zusammenhang der Theorie der absoluten optischen Instrumente mit einem Satz der Variationsrechnung, Münchener Sitzb. Math. -naturw Abteilung 1926 1–18; Ges. Math. Schr. II 181–197.
25. Georgiadou, Maria (2004). "5.29: Geometric Optics". Constantin Carathéodory: Mathematics and Politics in Turbulent Times. Germany: Springer. ISBN 3-540-20352-4.
26. Euler Opera Omnia, Series 1 (a) vol.24: Methodus inveniendi lineas curvas maximi minimive gaudentes sive solutio problematis isoperimetrici latissimo sensu accepti. Lausanne & Geneva 1744 (M. Bousquet) ed. C. Carathéodory Zürich 1952 (Fuesli). (b) vol.25 Commentationes analyticae ad calculum variationum pertinentes. ed C. Carathéodory Zürich 1952 (Fuesli).
27. Constantin Carathéodory: A Biography, newspaper article, 2000 "(...) Είχε γνωρίσει τον Ελευθέριο Βενιζέλο από το 1895, στην Κρήτη, και από το 1913 είχε προτείνει τη δημιουργία δεύτερου ελληνικού πανεπιστημίου στη Θεσσαλονίκη. Ο πόλεμος που ξεσπάει μεταθέτει τις αποφάσεις. Στην Ελλάδα θα επανέλθει το 1930-32, όταν θα αποδεχθεί τη θέση του κυβερνητικού επιτρόπου και θα οργανώσει τα πανεπιστήμια Αθήνας και Θεσσαλονίκης με τον νόμο 5343/32, ο οποίος ίσχυε μέχρι προσφάτως. Από τη θέση αυτή θα τον απολύσει η κυβέρνηση Παπαναστασίου που διαδέχεται τον Βενιζέλο το 1932 και εκεί θα σταματήσει η ενεργός ανάμειξή του στα κοινά της Ελλάδας." (Greek)
28. . Department of Primary Education, University of Patras. Archived from the original on 14 June 2012.
29. (PDF). National Technical University of Athens. Archived from the original (PDF) on 2017-12-22."His daughter Mrs Despina Rodopoulou – Carathéodory referred to this period: "He stayed to save anything he could: library, machines etc which were shipped in different ships hoping that one day they will arrive in Athens. My father stayed until the last moment. George Horton, consul of U.S.A. in Smyrni wrote a book... which was translated in Greek. In this book Horton notes: "One of the last Greek I saw on the streets of Smyrna before the entry of the Turks was Professor Carathéodory, president of the doomed University. With him departed the incarnation of Greek genious [sic] of culture and civilization on Orient." "
30. "Brief History". Aristotle University of Thessaloniki. Retrieved 2012-12-02.
31. Denker, Forscher und Entdecker: eine Geschichte der Bayerischen Akademie By Dietmar Willoweit p.263
32. Constantin Carathéodory-Hörsaal, mathe-lmu, Nr. 7/2002, Hrsg. Förderverein Mathematik in Wirtschaft, Universität und Schule an der Ludwig-Maximilians-Universität München e.V., S. 9.
33. (in Greek)"Caratheodory Museum Opening". Friends of C.Caratheodory.
34. . Hellenic Republic Embassy at Australia, Press and Communication Office. Archived from the original on 2010-01-04. Retrieved 2009-12-01.
35. "Caratheodory Museum enriched with new exhibits". Athens News Agency.
36. (in Greek). Eleftherotipia, major Greek newspaper. Archived from the original on 2011-10-02.
37. (in Greek). Kathimerini, major Greek newspaper. Archived from the original on 2011-07-16. Retrieved 2009-12-01.
38. (in Greek)"The museum of Carathéodory opened its gates to the public". Macedonia, Greek major newspaper.
39. Carathéodory, C. (1982). "Über die kanonischen Veränderlichen in der Variationsrechnung der mehrfachen Integrale". Festschrift zu seinem sechzigsten Geburtstag am 23.Januar 1922. Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 78–88. doi:10.1007/978-3-642-61810-9_11. ISBN 978-3-642-61810-9. S2CID 179177711.
40. Carathéodory, C. (1927). "Über das Schwarzsche Lemma bei analytischen Funktionen von zwei komplexen Veränderlichen". Mathematische Annalen. 97 (1): 76–98. doi:10.1007/BF01447861. S2CID 123411126.
41. Carathéodory, C. (1906). "Über die starken maxima und minima bei einfachen Integralen". Mathematische Annalen. 62 (4): 449–503. doi:10.1007/BF01449816. S2CID 115532504.
42. Carathéodory, C. (1909). "Untersuchungen Über die Grundlagen der Thermodynamik". Mathematische Annalen. 67 (3): 355–386. doi:10.1007/BF01450409. S2CID 118230148.
43. Carathéodory, C. Carathéodory (1914). "Elementarer Beweis für den Fundamentalsatz der konformen Abbildungen". Mathematische Abhandlungen Hermann Amandus Schwarz. Springer Berlin Heidelberg. pp. 19–41. doi:10.1007/978-3-642-50735-9_2. ISBN 978-3-642-50735-9.
44. Heins, Maurice (1951). "Review: Funktionentheorie by C. Carathéodory". Bulletin of the American Mathematical Society. 57 (3): 190–192. doi:10.1090/s0002-9904-1951-09486-0.

References

Books

• Maria Georgiadou, Constantin Carathéodory: Mathematics and Politics in Turbulent Times, Berlin-Heidelberg: Springer Verlag, 2004. ISBN 3-540-44258-8.
• Themistocles M. Rassias (editor) (1991) Constantin Caratheodory: An International Tribute, Teaneck, NJ: World Scientific Publishing Co., ISBN 981-02-0544-9.
• Nicolaos K. Artemiadis; translated by Nikolaos E. Sofronidis [2000](2004), History of Mathematics: From a Mathematician's Vantage Point, Rhode Island, USA: American Mathematical Society, pp. 270–4, 281, ISBN 0-8218-3403-7.
• Constantin Carathéodory in his...origins. International Congress at Vissa-Orestiada, Greece, 1–4 September 2000. Proceedings: T Vougiouklis (ed.), Hadronic Press, Palm Harbor FL 2001.

Biographical articles

• C. Carathéodory, Autobiographische Notizen, (In German) Wiener Akad. Wiss. 1954–57, vol.V, pp. 389–408. Reprinted in Carathéodory's Collected Writings vol.V. English translation in A. Shields, Carathéodory and conformal mapping, The Mathematical Intelligencer 10 (1) (1988), 18–22.
• O. Perron, Obituary: Constantin Carathéodory, Jahresberichte der Deutschen Mathematiker Vereinigung 55 (1952), 39–51.
• N. Sakellariou, Obituary: Constantin Carathéodory (Greek), Bull. Soc. Math. Grèce 26 (1952), 1–13.
• H Tietze, Obituary: Constantin Carathéodory, Arch. Math. 2 (1950), 241–245.
• H. Behnke, Carathéodorys Leben und Wirken, in A. Panayotopolos (ed.), Proceedings of C .Carathéodory International Symposium, September 1973, Athens (Athens, 1974), 17–33.
• Bulirsch R., Hardt M., (2000): Constantin Carathéodory: Life and Work, International Congress: "Constantin Carathéodory", 1–4 September 2000, Vissa, Orestiada, Greece

Encyclopaedias and reference works

• Chambers Biographical Dictionary (1997), Constantine Carathéodory, 6th ed., Edinburgh: Chambers Harrap Publishers Ltd, pp 270–1, ISBN 0-550-10051-2 (also available online).
• The New Encyclopædia Britannica (1992), Constantine Carathéodory, 15th ed., vol. 2, USA: The University of Chicago, Encyclopædia Britannica, Inc., pp 842, ISBN 0-85229-553-7 * New edition Online entry
• H. Boerner, Biography of Carathéodory in Dictionary of Scientific Biography (New York 1970–1990).

Conferences

• C. Carathéodory International Symposium, Athens, Greece September 1973. Proceedings edited by A. Panayiotopoulos (Greek Mathematical Society) 1975. Online
• Conference on Advances in Convex Analysis and Global Optimization (Honoring the memory of C. Carathéodory) June 5–9, 2000, Pythagorion, Samos, Greece. Online.
• International Congress: Carathéodory in his ... origins, September 1–4, 2000, Vissa Orestiada, Greece. Proceedings edited by Thomas Vougiouklis (Democritus University of Thrace), Hadronic Press FL USA, 2001. ISBN 1-57485-053-9.

External links

•   Media related to Constantin Caratheodory at Wikimedia Commons
• O'Connor, John J.; Robertson, Edmund F., "Constantin Carathéodory", MacTutor History of Mathematics Archive, University of St Andrews
• (in Greek) Web site dedicated to Carathéodory
• (in Greek) club www.s-karatheodoris.gr
• Constantin Carathéodory at the Mathematics Genealogy Project