Constantin Caratheodory
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Constantin Caratheodory

Constantin Carathéodory (Greek: , romanizedKonstantinos 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.

He was remembered by his colleagues as a respectable and cultured man.[3]

## 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, 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.[4] 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.[5]

## 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.

### 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.

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.[6]

Landau and Schwarz stimulated his interest in the study of complex analysis.[3]

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 G?ttingen, 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.[7]

## Works

### Calculus of variations

In his doctoral dissertation, Carathéodory showed how to extend solutions to discontinuous cases and studied isoperimetric problems.[3]

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. [8] 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.[8][9] 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.[3] 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.[8]

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.[8][10] The method can also be extended to multiple integrals.[]

### 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.[11]

He is credited with the authorship of the Carathéodory 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.

### 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.[12]

### Complex analysis

He greatly extended the theory of conformal transformation[13] 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.[3] He exhibited an elementary proof of the Schwarz lemma.[3]

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 holomphically equivalent to the bidisc.[3]

### 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.[14] In 1909, he published a pioneering work "Investigations on the Foundations of Thermodynamics"[15] 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.[16] Carathéodory's version attracted the attention of some of the top physicists of the time, including Max Planck, Max Born, and Arnold Sommerfeld.[3] According to Bailyn's survey of thermodynamics, Carathéodory's approach is called "mechanical," rather than "thermodynamic."[17] Max Born acclaimed this "first axiomatically rigid foundation of thermodynamics" and he expressed his enthusiasm in his letters to Einstein.[18][14] However, Max Planck had some misgivings[19] 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.[14]

In his theory he simplified the basic concepts, for instance heat is not an essential concept but a derived one. 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.[20]

### 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.[21] 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 it that 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.[22]

### 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.[23]

## The 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 [24] - the difficulties regarding the establishment of a Greek university in Constantinople led him to consider three other cities: Thessaloniki, Chios and Smyrna.[25]

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 rowing boat to the battleship Naxos which was standing by.[26] 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.[27]

## Linguistic and oratorical talents

Caratheodory at a mature age.

Carathéodory 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.[28] 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 extend 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, professor of ancient languages Kurt von Fritz praised Carathéodory, 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 numerous philosophical discussions with Carathéodory.

The Greek language was spoken exclusively in Carathéodory'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.

Carathéodory was a talented public speaker and was often invited to give speeches. In 1936, it was him who handed out the first ever Fields Medals at the meeting of the International Congress of Mathematicians in Oslo, Norway.[3]

## 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.[29]

In the town of Nea Vyssa, where Caratheodory's family came from, there is the unique Caratheodory's family museum. The museum is located in the central square of the town nearby the church and there are many personal items of Constantin as well as letters that he had exchanged with A. Einstein, for more information visit the original website of the club http://www.s-karatheodoris.gr. On the other hand, the Greek authorities intended for a long time to create a museum honoring Karatheodoris in Komotini, a major town of the northeastern Greek region, which is more than 200 km far away for the town of Nea Vyssa where his family came from. On 21 March 2009, the museum "Karatheodoris" () opened its gates to the public, in Komotini.[30][31][32]

The coordinator of the Museum, Athanasios Lipordezis ( ?), noted that the museum gave home to original manuscripts of the mathematician of about 10,000 pages including correspondence of Carathéodory with the German mathematician Arthur Rosenthal for the algebraization of measure. Also visitors can view at the showcases the books " Gesammelte mathematische Schriften Band 1,2,3,4 ", "Mass und ihre Algebraiserung", " Reelle Functionen Band 1", " Zahlen/Punktionen Funktionen " and many more. Handwritten letters of C.Carathéodory to Albert Einstein, Hellmuth Kneser and photographs of the Carathéodory family are on display.[]

Efforts to furnish the museum with more exhibits are ongoing.[33][34][35]

## 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[36]
• Über das Schwarzsche Lemma bei analytischen Funktionen von zwei komplexen Veränderlichen[37]
• 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.[38]
• Untersuchungen über die Grundlagen der Thermodynamik, Math. Ann. 67 (1909) pp. 355-386; Ges. Math. Schr. II 131-166.[39]
• 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.[40]
• 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
• Functionentheorie I, II, Basel 1950,[41] 1961 (Birkhäuser). English translation: Theory of Functions of a Complex Variable, 2 vols, New York, Chelsea Publishing Company, 1954
• 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.

## Notes

1. ^ "The Mathematics Genealogy Project - Constantin Carathéodory". Mathematics Genealogy Project. North Dakota State University Department of Mathematics. Archived from the original on 13 July 2018. Retrieved 2017.
2. ^ "The Mathematics Genealogy Project - Naz?m Terzio?lu". Mathematics Genealogy Project. North Dakota State University Department of Mathematics. Retrieved 2017.
3. 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.
4. ^ Brussells 1901 (Hayez);Ges. math. Schr. V. 273-281
5. ^ H Aigyptos, Syllogos Ophelimon Biblion, no 14, 118 pp Athens 1901, 1928, New York 1920
6. ^ Georgiadou, Maria (2004). "2.15: Einstein Contacts Carathéodory". Constantin Carathéodory: Mathematics and Politics in Turbulent Times. Germany: Springer. ISBN 3-540-20352-4.
7. ^ J P Christianidis & N Kastanis: In memoriam Evangelos S Stamatis (1898-1990) Historia Mathematica 19 (1992) 99-105
8. ^ a b c d Kot, Mark (2014). "Chapter 12: Sufficient Conditions". A First Course in the Calculus of Variations. American Mathematical Society. ISBN 978-1-4704-1495-5.
9. ^ H. Boerner, Carathéodory und die Variationsrechnung, in A Panayotopolos (ed.), Proceedings of C. Carathéodory International Symposium, September 1973, Athens (Athens, 1974), 80-90.
10. ^ 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
11. ^ 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.
12. ^ Bartle, Robert G.; Sherbert, Donald R. (2011). "6.1: The Derivative". Introduction to Real Analysis. John Wiley & Sons. ISBN 978-0-471-43331-6.
13. ^ A. Shields: Carathéodory and Conformal Mapping Math. Intelligencer vol.10(1), 1988
14. ^ a b c Georgiadou, Maria (2004). "2.2 Axiomatic Foundation of Thermodynamics". Constantin Carathéodory: Mathematics and Politics in Turbulent Times. Germany: Springer. ISBN 3-540-20352-4.
15. ^ Carathéodory, Constantin (1909). Translated by Delphinich, D. H. "Untersuchungen ueber die Grundlagen der Thermodynamik" [Examination of the foundations of Thermodynamics] (PDF). Mathematische Annalen. 67 (3): 355-386. doi:10.1007/bf01450409. Archived from the original (PDF) on 2019-10-12. Retrieved .
16. ^ 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.
17. ^ Bailyn, M. (1994). A Survey of Thermodynamics, American Institute of Physics, Woodbury NY, ISBN 0-88318-797-3.
18. ^ Max Born: The Born-Einstein Letters, MacMillan 1971
19. ^ Constantin Carathéodory and the axiomatic thermodynamics by Lionello Pogliani and Mario N. Berberan-Santos
20. ^ 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
21. ^ 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.
22. ^ Georgiadou, Maria (2004). "5.29: Geometric Optics". Constantin Carathéodory: Mathematics and Politics in Turbulent Times. Germany: Springer. ISBN 3-540-20352-4.
23. ^ 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).
24. ^ Constantin Carathéodory: A Biography, newspaper article, 2000 "(...) ? 1895, ? , 1913 ? ? ? . ? ? ? . ? 1930-32, ? ? ? 5343/32, ? . ? ? ? ? ? 1932 ? ? ? ? ?." (Greek)
25. ^ "The importance of the foundation of the University of Smyrna(Essay)". Department of Primary Education, University of Patras. Archived from the original on 14 June 2012.
26. ^ "Constantin Carathéodory: His life and work(Essay)" (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 of culture and civilization on Orient." "
27. ^ "Brief History". Aristotle University of Thessaloniki. Retrieved .
28. ^ Denker, Forscher und Entdecker: eine Geschichte der Bayerischen Akademie By Dietmar Willoweit p.263
29. ^ 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.
30. ^ (in Greek)"Caratheodory Museum Opening". Friends of C.Caratheodory.
31. ^ "Caratheodory Museum Opens". Hellenic Republic Embassy at Australia, Press and Communication Office. Archived from the original on 2010-01-04. Retrieved .
32. ^ "Caratheodory Museum enriched with new exhibits". Athens News Agency.
33. ^ (in Greek)"The museum of C.Carathéodory at Komotini". Eleftherotipia, major Greek newspaper. Archived from the original on 2011-10-02.
34. ^ (in Greek)"Carathéodory Museum: attractor". Kathimerini, major Greek newspaper. Archived from the original on 2011-07-16. Retrieved .
35. ^ (in Greek)"The museum of Carathéodory opened its gates to the public". Macedonia, Greek major newspaper.
36. ^ 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.
37. ^ Carathéodory, C. (1927). "Über das Schwarzsche Lemma bei analytischen Funktionen von zwei komplexen Veränderlichen". Mathematische Annalen. 97 (1): 76-98. doi:10.1007/BF01447861.
38. ^ Carathéodory, C. (1906). "Über die starken maxima und minima bei einfachen Integralen". Mathematische Annalen. 62 (4): 449-503. doi:10.1007/BF01449816.
39. ^ Carathéodory, C. (1909). "Untersuchungen Über die Grundlagen der Thermodynamik" (PDF). Mathematische Annalen. 67 (3): 355-386. doi:10.1007/BF01450409.
40. ^ 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.
41. ^ 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.