View source for Constantin Carathéodory

{{Short description|Greek mathematician (1873–1950)}}
{{for|the Ottoman Greek doctor|Constantine Caratheodory (1802-1879)}}
{{Infobox scientist
| name              = Constantin Carathéodory
| image             = Caratheodory (cropped).jpg
| image_size        = 
| caption           = Constantin Carathéodory
| birth_date        = {{birth date|1873|9|13|df=y}}
| birth_place       = [[Berlin]], [[German Empire]]
| death_date        = {{death date and age|1950|2|2|1873|9|13|df=y}}
| death_place       = [[Munich]], [[West Germany]]
| nationality       = [[Greece|Greek]]
| field             = [[Calculus of variations]] <br/> [[Real analysis]] <br/> [[Complex analysis]] <br/> [[Measure theory]]
| work_institution  = *[[University of Bonn]]
*[[University of Hannover|Hannover Technical High School]]
*[[Wrocław University of Technology|Breslau Technical High School]]
*[[University of Göttingen]]
*[[University of Berlin]]
*[[University of Munich]]
*[[National Technical University of Athens]]
*[[Ionian University of Smyrna]]
| alma_mater        = [[University of Berlin]]<br>[[University of Göttingen]]
| doctoral_advisor  = [[Hermann Minkowski]]<ref name=mathgen1>{{cite web|url=http://genealogy.math.ndsu.nodak.edu/id.php?id=7517|website=Mathematics Genealogy Project|publisher=North Dakota State University Department of Mathematics|access-date=27 August 2017|title=The Mathematics Genealogy Project - Constantin Carathéodory|archive-url=https://web.archive.org/web/20180713153220/https://www.genealogy.math.ndsu.nodak.edu/id.php?id=7517|archive-date=13 July 2018|url-status=dead}}</ref>
| doctoral_students = [[Paul Finsler]]<br>[[Hans Rademacher]]<br>[[Georg Aumann]]<br>[[Hermann Boerner]]<br>[[Ernst Peschl]]<br>[[Wladimir Seidel]]<br>[[Nazim Terzioglu|Nazım Terzioğlu]]<ref name=mathgen2>{{cite web|url=http://genealogy.math.ndsu.nodak.edu/id.php?id=148430|website=Mathematics Genealogy Project|publisher=North Dakota State University Department of Mathematics|access-date=27 August 2017|title=The Mathematics Genealogy Project - Nazım Terzioğlu}}</ref><br>[[Xu Ruiyun]]
| known_for         = [[Carathéodory conjecture]]<br/>[[Carathéodory function]]<br/>[[Carathéodory metric]]<br/>[[Carathéodory's theorem (disambiguation)|Carathéodory theorems]]<br>[[Carathéodory's criterion]]<br/>[[Carathéodory's lemma]]<br/>[[Positive harmonic function#Carathéodory's positivity criterion for holomorphic functions|Carathéodory's positivity criterion for holomorphic functions]]<br/>[[Second law of thermodynamics#Principle of Carathéodory|Carathéodory's principle]]<br/>[[Sub-Riemannian manifold|Carnot–Carathéodory metric]]<br/>[[Adiabatic accessibility]]<br/>[[Cyclic polytope]]<br/>[[Prime end]]<br/> General theory of [[outer measure]]s<br>Axiomatic formulation of [[thermodynamics]]
| signature = Caratheodory signature.png
}}
'''Constantin Carathéodory''' ({{langx|el|Κωνσταντίνος Καραθεοδωρή|Konstantinos Karatheodori}}; 13 September 1873 – 2 February 1950) was a [[Greeks|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<ref>{{Cite book|last1=Hallett|first1=Michael|url=https://books.google.com/books?id=NxENTTHJCz4C|title=David Hilbert's Lectures on the Foundations of Geometry 1891–1902|last2=Majer|first2=Ulrich|date=2004|publisher=Springer Science & Business Media|isbn=978-3-540-64373-9|pages=11|language=en}}</ref> and the most renowned [[Greek mathematics|Greek mathematician]] since [[Ancient history|antiquity]].<ref>{{Cite book|last=Szpiro|first=George G.|url=https://books.google.com/books?id=iEBOce-4S2EC|title=Poincare's Prize: The Hundred-Year Quest to Solve One of Math's Greatest Puzzles|date=2008|publisher=Penguin|isbn=978-1-4406-3428-4|pages=104|language=en}}</ref>

== Origins ==
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| image1 = Caratheodory Constantin and Stephanos.JPG
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| caption1 = Carathéodory with his father,  Stephanos, in 1900.
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| caption2 = Carathéodory (left) pictured sitting with his father, brother in law and sister, Carlsbad 1898
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Constantin Carathéodory was born in 1873 in [[Berlin]] to [[Greeks|Greek]] parents and grew up in [[Brussels]]. His father {{Interlanguage link|Stephanos Karatheodori|lt=Stephanos|tr|İstefanaki Karatodori (diplomat)}}, a lawyer, served as the [[Ottoman Empire|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 [[Istanbul|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 [[Greco-Turkish War (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 [[Camille Jordan|Jordan's]] ''Cours d'Analyse'' and [[George Salmon|Salmon's]] text on the analytic geometry of [[conic sections]]. He also visited the [[Pyramid of Cheops|Cheops pyramid]] and made measurements which he wrote up and published in 1901.<ref>Brussells 1901 (Hayez);Ges. math. Schr. V. 273-281</ref> 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.<ref>H Aigyptos, Syllogos Ophelimon Biblion, no 14, 118 pp Athens 1901, 1928, New York 1920</ref>

{{Clear}}

== Studies and university career ==
[[File:Caratheodory Constantine.JPG|thumb|Young Carathéodory]]
Carathéodory studied engineering in [[Belgium]] at the [[Royal Military Academy (Belgium)|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 [[University of Bonn|Bonn]]
* 1909  Ordinary Professor at [[University of Hannover|Hannover Technical High School]].
* 1910  Ordinary Professor at [[Wrocław University of Technology|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 at 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===
[[Image:Caratheodory and Fejér.JPG|right|thumb|200px|Carathéodory (left) with Hungarian mathematician Lipót Fejér (1880–1959) (standing to the right).]]Carathéodory had numerous contacts in Germany. They included such famous names as: [[Hermann Minkowski]], [[David Hilbert]], [[Felix Klein]], [[Albert Einstein]], [[Edmund Landau]], [[Hermann Amandus Schwarz]], and [[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 a member of the Prussian Academy of Sciences in Berlin, was working on his general theory of relativity when he contacted Carathéodory for clarifications on the [[Hamilton–Jacobi equation|Hamilton-Jacobi equation]] and [[canonical transformation]]s. 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|E.T. Whittaker's]] ''[[Analytical Dynamics of Particles and Rigid Bodies|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.<ref>{{Cite book|last=Georgiadou|first=Maria|title=Constantin Carathéodory: Mathematics and Politics in Turbulent Times|publisher=Springer|year=2004|isbn=3-540-20352-4|location=Germany|chapter=2.15: Einstein Contacts Carathéodory}}</ref>

Landau and Schwarz stimulated his interest in the study of complex analysis.<ref name=":0">{{Cite book|last=Begehr|first=H. G. W.|title=Mathematics in Berlin|publisher=Birkhäuser Verlag|year=1998|isbn=3-7643-5943-9|editor-last=Begehr, H. G. W.|location=Germany|chapter=Constantin Carathéodory (1873-1950)|editor-last2=Koch, H|editor-last3=Krammer, J|editor-last4=Schappacher, N|editor-last5=Thiele, E.-J}}</ref>

===Academic contacts in Greece===
While in Germany, Carathéodory retained numerous links with the Greek academic world, details of which can 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 at Athens University, Carathéodory had Evangelos Stamatis as an undergraduate student, who subsequently achieved considerable distinction as a scholar of ancient Greek mathematical classics.<ref>J P Christianidis & N Kastanis: ''In memoriam Evangelos S Stamatis (1898–1990) Historia Mathematica 19 (1992) 99-105''</ref>

== Works ==
===Calculus of variations===
In his doctoral dissertation, Carathéodory showed how to extend solutions to discontinuous cases and studied isoperimetric problems.<ref name=":0" />

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.<ref name=":2">{{Cite book|last=Kot|first=Mark|title=A First Course in the Calculus of Variations|publisher=American Mathematical Society|year=2014|isbn=978-1-4704-1495-5|chapter=Chapter 12: Sufficient Conditions}}</ref> 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''.<ref name=":2" /><ref>H. Boerner, ''Carathéodory und die Variationsrechnung'', in A Panayotopolos (ed.), Proceedings of C. Carathéodory International Symposium, September 1973, Athens (Athens, 1974), 80–90.</ref> 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.<ref name=":0" /> It allows for quick and elegant derivations of conditions of sufficiency in the calculus of variations and leads directly to the [[Euler–Lagrange equation|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.<ref name=":2" />

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.<ref name=":2" /><ref>[[Richard Bellman|Bellman]] for his ''Dynamic programming'' in its continuous-time form used Carathéodory's work in the form of the [[Hamilton–Jacobi–Bellman equation]]. [[Rudolf E. Kálmán|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</ref>

===Convex geometry===
[[File:Caratheodorys theorem example.png|thumb|200px|An illustration of [[Carathéodory's theorem (convex hull)]] for a square in '''R'''<sup>2</sup>.]][[Carathéodory's theorem (convex hull)|Carathéodory's theorem]] in convex geometry states that if a point <math>x</math> of <math>\mathbb{R}^d</math> lies in the [[convex hull]] of a set <math>P</math>, then <math>x</math> can be written as the convex combination of at most <math>d + 1</math> points in <math>P</math>. Namely, there is a subset <math>P'</math> of <math>P</math> consisting of <math>d + 1</math> or fewer points such that <math>x</math> lies in the convex hull of <math>P'</math>. Equivalently, <math>x</math> lies in an <math>r</math>-[[simplex]] with vertices in <math>P</math>, where <math>r \leq  d</math>.  The smallest <math>r</math> that makes the last statement valid for each <math>x</math> in the convex hull of ''P'' is defined as the ''Carathéodory's number'' of <math>P</math>. Depending on the properties of <math>P</math>, upper bounds lower than the one provided by Carathéodory's theorem can be obtained.<ref>{{Cite journal|last1=Bárány|first1=Imre|last2=Karasev|first2=Roman|date=2012-07-20|title=Notes About the Carathéodory Number|journal=Discrete & Computational Geometry|language=en|volume=48|issue=3|pages=783–792|arxiv=1112.5942|doi=10.1007/s00454-012-9439-z|s2cid=9090617|issn=0179-5376}}</ref>

He is credited with the authorship of the [[Carathéodory conjecture]] claiming that a closed convex surface admits at least two [[umbilic point]]s. The conjecture was proven in 2024 by Brendan Guilfoyle and Wilhelm Klingenberg. <ref name="GuilKling19">{{cite journal |first1=B. |last1=Guilfoyle |first2=W. |last2=Klingenberg |year=2019 |title= Higher codimensional mean curvature flow of compact spacelike submanifolds |journal= Trans. Amer. Math. Soc.|volume=372 |issue=9 |pages=6263–6281 |doi=10.1090/tran/7766|s2cid=119253397 |doi-access=free |arxiv=1812.00710 }}</ref> <ref name="GuilKling20">{{cite journal |first1=B. |last1=Guilfoyle |first2=W. |last2=Klingenberg |year=2020 |title= Fredholm-regularity of holomorphic discs in plane bundles over compact surfaces |journal= Ann. Fac. Sci. Toulouse Math. |series=Série 6 |volume=29 |issue=3 |pages=565–576 | arxiv=1812.00707 | doi=10.5802/afst.1639|s2cid=119659239 |doi-access=free }}</ref> <ref name="GuilKling24">{{cite journal |first1=B. |last1=Guilfoyle|first2=W. |last2=Klingenberg |title=Proof of the Toponogov Conjecture on complete surfaces |journal=J. Gökova Geom. Topol. GGT |volume=17 | year=2024 | pages=1–50| arxiv=2002.12787| url=https://gokovagt.org/journal/2024/guilfoyleklingenberg.html}}</ref>

===Real analysis===
He proved an [[Carathéodory's existence theorem|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|Chain Rule]] and the formula for the [[Inverse functions and differentiation|derivative of inverse functions]].<ref>{{Cite book|last1=Bartle|first1=Robert G.|title=Introduction to Real Analysis|last2=Sherbert|first2=Donald R.|publisher=John Wiley & Sons|year=2011|isbn=978-0-471-43331-6|chapter=6.1: The Derivative}}</ref>

===Complex analysis===
He greatly extended the theory of [[conformal transformation]]<ref>A. Shields: ''Carathéodory and Conformal Mapping'' Math. Intelligencer vol.10(1), 1988</ref> proving his [[Carathéodory's theorem (conformal mapping)|theorem]] about the extension of conformal mapping to the boundary of Jordan domains. In studying boundary correspondence he originated the theory of [[prime end]]s.<ref name=":0" /> He exhibited an elementary proof of the [[Schwarz lemma]].<ref name=":0" />

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 <math>\mathbb{C}^2</math> is not holomorphically equivalent to the bidisc.<ref name=":0" />

===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 algebra (structure)|Boolean algebra]]s.

===Thermodynamics===
Thermodynamics had been a subject dear to Carathéodory since his time in Belgium.<ref name=":1">{{Cite book|last=Georgiadou|first=Maria|title=Constantin Carathéodory: Mathematics and Politics in Turbulent Times|publisher=Springer|year=2004|isbn=3-540-20352-4|location=Germany|chapter=2.2 Axiomatic Foundation of Thermodynamics}}</ref> In 1909, he published a pioneering work "Investigations on the Foundations of Thermodynamics"<ref>{{cite journal|last=Carathéodory|first=Constantin|title=Untersuchungen ueber die Grundlagen der Thermodynamik|trans-title=Examination of the foundations of Thermodynamics|journal=[[Mathematische Annalen]]|volume=67|issue=3|year=1909|pages=355–386|doi=10.1007/bf01450409|s2cid=118230148|translator-last=Delphinich|translator-first=D. H.|url=http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf|access-date=2016-07-09|archive-url=https://web.archive.org/web/20191012152205/http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf|archive-date=2019-10-12|url-status=dead}}</ref> 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 theorem|Clausius]], and of [[Kelvin–Planck statement|Kelvin and Planck]].<ref>{{Cite book|last=Lewis|first=Christopher J. T.|title=Heat and Thermodynamics: A Historical Perspective|publisher=Greenwood Press|year=2007|isbn=978-0-313-33332-3|location=Westport, Connecticut|pages=110|chapter=Chapter 5. Energy and Entropy: The Birth of Thermodynamics.}}</ref> Carathéodory's version attracted the attention of some of the top physicists of the time, including Max Planck, Max Born, and Arnold Sommerfeld.<ref name=":0" /> According to  Bailyn's survey of thermodynamics, Carathéodory's approach is called "mechanical," rather than "thermodynamic."<ref>Bailyn, M. (1994). ''A Survey of Thermodynamics'', American Institute of Physics, Woodbury NY, {{isbn|0-88318-797-3}}.</ref> Max Born acclaimed this "first axiomatically rigid foundation of thermodynamics" and he expressed his enthusiasm in his letters to Einstein.<ref>Max Born: ''The Born–Einstein Letters'', MacMillan 1971</ref><ref name=":1" /> However, Max Planck had some misgivings<ref>''[http://web.ist.utl.pt/ist12219/data/68.pdf Constantin Carathéodory and the axiomatic thermodynamics]'' by Lionello Pogliani and Mario N. Berberan-Santos</ref> 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.<ref name=":1" />

In his theory he simplified the basic concepts, for instance ''heat'' is not an essential concept but a derived one.<ref>{{Cite journal |last1=Pogliani |first1=Lionello |last2=Berberan-Santos |first2=Mario N. |date=2000 |title=Constantin Carathéodory and the axiomatic thermodynamics |url=http://link.springer.com/10.1023/A:1018834326958 |journal=Journal of Mathematical Chemistry |volume=28 |issue=1/3 |pages=313–324 |doi=10.1023/A:1018834326958|s2cid=17244147 }}</ref> 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]].<ref>adiabatic accessibility = [[:de:Adiabatische Erreichbarkeit|adiabatische Erreichbarkeit]]; see also Elliott H. Lieb, Jakob Yngvason: [http://www.arxiv.org/abs/cond-mat/9708200 ''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}}</ref>

===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 in optical systems|aberration]], except for the trivial case of plane mirrors.
In his later work he gave the theory of the [[Schmidt telescope]].<ref>''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.</ref> 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|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.<ref>{{Cite book|last=Georgiadou|first=Maria|title=Constantin Carathéodory: Mathematics and Politics in Turbulent Times|publisher=Springer|year=2004|isbn=3-540-20352-4|location=Germany|chapter=5.29: Geometric Optics}}</ref>

===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.<ref>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).</ref>

== The University of Smyrna ==
[[File:Ionian University of Smyrna.jpg|thumb|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 needs of the eastern part of the Aegean Sea and the [[Balkans]]. Carathéodory, who was a professor at the [[University of Berlin]] at the time, proposed the establishment of a new university <ref>[http://www.tovima.gr/books-ideas/article/?aid=127956 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)]</ref> - the difficulties regarding the establishment of a Greek university in [[Constantinople]] led him to consider three other cities: [[Thessaloniki]], [[Chios]] and [[Smyrna]].<ref name="The importance of the foundation of the University of Smyrni">{{cite web|url=http://www.elemedu.upatras.gr/eriande/synedria/synedrio4/praktika1/pyrgiotakis.htm|title=The importance of the foundation of the University of Smyrna(Essay)|publisher=Department of Primary Education, University of Patras|archive-url=https://web.archive.org/web/20120614082453/http://www.elemedu.upatras.gr/eriande/synedria/synedrio4/praktika1/pyrgiotakis.htm|archive-date=14 June 2012}}</ref>

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 [[Greco-Turkish War (1919–1922)|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.<ref name="Constantin Carathéodory: His life and work">{{cite web|url=http://www.24grammata.com/wp-content/uploads/2011/12/Caratheodori-24grammata.com_.pdf |title=Constantin Carathéodory: His life and work(Essay)|publisher=National Technical University of Athens |archive-url=https://web.archive.org/web/20171222123424/http://www.24grammata.com/wp-content/uploads/2011/12/Caratheodori-24grammata.com_.pdf |archive-date=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 {{sic|gen|ious}} of culture and civilization on Orient." ''"</ref> Carathéodory brought to Athens some of the university library and stayed there, 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 at 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.<ref name="Brief History of Aristotle University of Thessaloniki">{{cite web|url=https://www.auth.gr/en/history|title=Brief History|publisher=Aristotle University of Thessaloniki|access-date=2012-12-02}}</ref>

== Linguistic and oratorical talents ==
[[Image:Caratheodory constantin.jpg|thumb|200px|Caratheodory at a mature age.|alt=]]Carathéodory excelled at languages, much like many members of his family. [[Greek language|Greek]] and [[French language|French]] were his first languages, and he mastered [[German language|German]] with such perfection, that his writings composed in the German language are stylistic masterworks.<ref>Denker, Forscher und Entdecker: eine Geschichte der Bayerischen Akademie
 By Dietmar Willoweit p.263</ref> Carathéodory also spoke and wrote [[English language|English]], [[Italian language|Italian]], [[Turkish language|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 [[philology|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 Medal]]s at the meeting of the International Congress of Mathematicians in Oslo, Norway.<ref name=":0" />

{{Clear}}

== Legacy ==
[[File:Grave Caratheodory 2019b.jpg|thumb|253x253px|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.<ref>[http://www.mathematik.uni-muenchen.de/~fmwus/download/ausgabe07.pdf 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.</ref>

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 200&nbsp;km away from his home town above. On 21 March 2009, the "Karatheodoris" Museum (Καραθεοδωρής) opened its gates to the public in Komotini.<ref>{{in lang|el}}{{cite web|url=http://www.karatheodori.gr/index.php?op=news&lop=viewNew&nid=20|title=Caratheodory Museum Opening|publisher= Friends of C.Caratheodory}}</ref><ref>{{cite web|url=http://www.greekembassy.org.au/media_news.php?act=detail&id=267|title=Caratheodory Museum Opens|publisher=Hellenic Republic Embassy at Australia, Press and Communication Office|access-date=2009-12-01|archive-url=https://web.archive.org/web/20100104215737/http://www.greekembassy.org.au/media_news.php?act=detail&id=267|archive-date=2010-01-04|url-status=dead}}</ref><ref>{{cite web|url=http://www.hri.org/news/greek/ana/2009/09-03-20.ana.html#36|title=Caratheodory Museum enriched with new exhibits|publisher=Athens News Agency}}</ref>

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.<ref>{{in lang|el}}{{cite web|url=http://archive.enet.gr/online/online_text/c=112,dt=23.03.2009,id=53237924|title=The museum of C.Carathéodory at Komotini|publisher=Eleftherotipia, major Greek newspaper|url-status=dead|archive-url=https://web.archive.org/web/20111002122646/http://archive.enet.gr/online/online_text/c=112,dt=23.03.2009,id=53237924|archive-date=2011-10-02}}</ref><ref>{{in lang|el}}{{cite web|url=http://portal.kathimerini.gr/4dcgi/_w_articles_kathextra_1_02/04/2009_273714|title=Carathéodory Museum: attractor|publisher=Kathimerini, major Greek newspaper|access-date=2009-12-01|archive-url=https://web.archive.org/web/20110716055345/http://portal.kathimerini.gr/4dcgi/_w_articles_kathextra_1_02/04/2009_273714|archive-date=2011-07-16|url-status=dead}}</ref><ref>{{in lang|el}}{{cite web|url=http://www.makthes.gr/index.php?name=News&file=article&sid=35863|title=The museum of Carathéodory opened its gates to the public|publisher=Macedonia, Greek major newspaper}}</ref>

== 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''<ref>{{cite book|title=Festschrift zu seinem sechzigsten Geburtstag am 23.Januar 1922|date=1982|publisher=Springer Berlin Heidelberg|location=Berlin, Heidelberg|isbn=978-3-642-61810-9|pages=78–88|doi=10.1007/978-3-642-61810-9_11|chapter=Über die kanonischen Veränderlichen in der Variationsrechnung der mehrfachen Integrale|last1=Carathéodory|first1=C.|s2cid=179177711 }}</ref>
* ''Über das Schwarzsche Lemma bei analytischen Funktionen von zwei komplexen Veränderlichen''<ref>{{cite journal|last=Carathéodory|first=C.|title=Über das Schwarzsche Lemma bei analytischen Funktionen von zwei komplexen Veränderlichen|journal=Mathematische Annalen|volume=97|issue=1|pages=76–98|doi=10.1007/BF01447861|year=1927|s2cid=123411126}}</ref>
* ''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.<ref>{{cite journal|last=Carathéodory|first=C.|title=Über die starken maxima und minima bei einfachen Integralen|journal=Mathematische Annalen|volume=62|issue=4|pages=449–503|doi=10.1007/BF01449816|year=1906|s2cid=115532504|url=https://zenodo.org/record/1585874}}</ref>
* ''Untersuchungen über die Grundlagen der Thermodynamik'', Math. Ann. 67 (1909) pp.&nbsp;355–386; Ges. Math. Schr. II 131–166.<ref>{{cite journal|last=Carathéodory|first=C.|title=Untersuchungen Über die Grundlagen der Thermodynamik|journal=Mathematische Annalen|volume=67|issue=3|pages=355–386|doi=10.1007/BF01450409|year=1909|s2cid=118230148|url=https://zenodo.org/record/1428268}}</ref>
* ''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.<ref>{{cite book|last=Carathéodory|first=C.|title=Mathematische Abhandlungen Hermann Amandus Schwarz|date=1914|publisher=Springer Berlin Heidelberg|isbn=978-3-642-50735-9|pages=19–41|doi=10.1007/978-3-642-50735-9_2|chapter=Elementarer Beweis für den Fundamentalsatz der konformen Abbildungen}}</ref>
* ''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 ===
* {{Citation | last1=Carathéodory | first1=Constantin | title=Vorlesungen über reelle Funktionen | url=https://books.google.com/books?id=9rJXAAAAYAAJ | publisher=Teubner | location=Leipzig | edition=3rd | isbn=978-0-8284-0038-1 | mr=0225940  | year=1918}} 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,<ref>{{cite journal|url=http://projecteuclid.org/download/pdf_1/euclid.bams/1183516047|author=Heins, Maurice|author-link=Maurice Heins|title=Review: ''Funktionentheorie'' by C. Carathéodory|journal=Bulletin of the American Mathematical Society|volume=57|issue=3|year=1951|pages=190–192|doi=10.1090/s0002-9904-1951-09486-0|doi-access=free}}</ref> 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==
{{Portal|Biography|Mathematics|Physics}}
* [[Domain (mathematical analysis)]]
* [[Nemytskii operator]]
* [[Herbert Callen]], who also sought an axiomatic formulation of thermodynamics

== Notes ==
{{Reflist|30em}}

== References ==

=== Books ===
* Maria Georgiadou, ''[https://books.google.com/books?id=IVIXBOFNty8C 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.&nbsp;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.&nbsp;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.
* [[Oskar Perron|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.
* [[Heinrich Tietze|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  [http://www.chambersharrap.co.uk/chambers/features/chref/chref.py/main?query=caratheodory&title=biog 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}} * [http://www.britannica.com/eb/article-9020226/Constantin-Caratheodory 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. [http://www.hms.gr/apothema/?s=scf&i=29 Online]
* Conference on ''Advances in Convex Analysis and Global Optimization (Honoring the memory of C. Carathéodory)'' June 5–9, 2000, Pythagorion, Samos, Greece. [http://www.samos.aegean.gr/math/acago 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 ==
*{{Commons category-inline}}
* {{MacTutor Biography|id=Caratheodory}}
* {{in lang|el}} [http://www.karatheodori.gr/ Web site dedicated to Carathéodory]
* {{in lang|el}} [http://www.s-karatheodoris.gr  club www.s-karatheodoris.gr]
* {{MathGenealogy |id=7517}}

{{Authority control}}

{{DEFAULTSORT:Caratheodory, Constantin}}
[[Category:1873 births]]
[[Category:1950 deaths]]
[[Category:20th-century German mathematicians]]
[[Category:Greeks from the Ottoman Empire]]
[[Category:German people of Greek descent]]
[[Category:19th-century Greek mathematicians]]
[[Category:Eastern Orthodox Christians from Germany]]
[[Category:Complex analysts]]
[[Category:Mathematical analysts]]
[[Category:Members of the Prussian Academy of Sciences]]
[[Category:Thermodynamicists]]
[[Category:Scientists from Berlin]]
[[Category:Scientists from Brussels]]
[[Category:Occupation of Smyrna]]
[[Category:University of Göttingen alumni]]
[[Category:Members of the Academy of Athens (modern)]]
[[Category:Variational analysts]]
[[Category:19th-century Greek scientists]]
[[Category:Measure theorists]]
[[Category:Emigrants from the Ottoman Empire to Germany]]
[[Category:People of the Burning of Smyrna]]