Moritz Werner Fenchel (; 3 May 1905 – 24 January 1988) was a
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
known for his contributions to
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
and to
optimization theory
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
. Fenchel established the basic results of
convex analysis
Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory.
Convex sets
A subset C \subseteq X of som ...
and nonlinear optimization theory which would, in time, serve as the foundation for
nonlinear programming
In mathematics, nonlinear programming (NLP) is the process of solving an optimization problem where some of the constraints or the objective function are nonlinear. An optimization problem is one of calculation of the extrema (maxima, minima or s ...
. A German-born Jew and early refugee from Nazi suppression of intellectuals, Fenchel lived most of his life in Denmark. Fenchel's monographs and lecture notes are considered influential.
Biography
Early life and education
Fenchel was born on 3 May 1905 in
Berlin
Berlin ( , ) is the capital and largest city of Germany by both area and population. Its 3.7 million inhabitants make it the European Union's most populous city, according to population within city limits. One of Germany's sixteen constitue ...
, Germany,
his younger brother was the Israeli film director and architect
Heinz Fenchel
Heinz Fenchel (1906–1988) was a German art director, who designed the sets for a number of films in Weimar Germany and Czechoslovakia. He worked on several films with the director Pál Fejös.Büttner p.231-33 The Jewish Fenchel later emigrated ...
.
Fenchel studied mathematics and physics at the
University of Berlin
Humboldt-Universität zu Berlin (german: Humboldt-Universität zu Berlin, abbreviated HU Berlin) is a German public research university in the central borough of Mitte in Berlin. It was established by Frederick William III on the initiative o ...
between 1923 and 1928.
He wrote his doctorate thesis in
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
(''Über Krümmung und Windung geschlossener Raumkurven'') under
Ludwig Bieberbach
Ludwig Georg Elias Moses Bieberbach (; 4 December 1886 – 1 September 1982) was a German mathematician and Nazi.
Biography
Born in Goddelau, near Darmstadt, he studied at Heidelberg and under Felix Klein at Göttingen, receiving his doctorat ...
.
Professorship in Germany
From 1928 to 1933, Fenchel was Professor
E. Landau's Assistant at the
University of Göttingen
The University of Göttingen, officially the Georg August University of Göttingen, (german: Georg-August-Universität Göttingen, known informally as Georgia Augusta) is a public research university in the city of Göttingen, Germany. Founded ...
. During a one-year leave (on
Rockefeller Fellowship
The Rockefeller Foundation is an American private foundation and philanthropic medical research and arts funding organization based at 420 Fifth Avenue, New York City. The second-oldest major philanthropic institution in America, after the Ca ...
) between 1930 and 1931, Fenchel spent time in Rome with
Levi-Civita
Tullio Levi-Civita, (, ; 29 March 1873 – 29 December 1941) was an Italian mathematician, most famous for his work on absolute differential calculus (tensor calculus) and its applications to the theory of relativity, but who also made signific ...
, as well as in
Copenhagen
Copenhagen ( or .; da, København ) is the capital and most populous city of Denmark, with a proper population of around 815.000 in the last quarter of 2022; and some 1.370,000 in the urban area; and the wider Copenhagen metropolitan ar ...
with
Harald Bohr
Harald August Bohr (22 April 1887 – 22 January 1951) was a Danish mathematician and footballer. After receiving his doctorate in 1910, Bohr became an eminent mathematician, founding the field of almost periodic functions. His brother was the No ...
and
Tommy Bonnesen
Tommy Bonnesen (27 March 1873 – 14 March 1935) was a Danish mathematician, known for Bonnesen's inequality.
Bonnesen studied at the University of Copenhagen, where in 1902 he received his Ph.D. (promotion) with thesis ''Analytiske studier over ...
.
He visited Denmark again in 1932.
Professorship in exile
Fenchel taught at Göttingen until 1933, when the
Nazi discrimination laws led to
mass-firings of Jews.
Fenchel emigrated to Denmark somewhere between April and September 1933, ultimately obtaining a position at the
University of Copenhagen
The University of Copenhagen ( da, Københavns Universitet, KU) is a prestigious public university, public research university in Copenhagen, Copenhagen, Denmark. Founded in 1479, the University of Copenhagen is the second-oldest university in ...
. In December 1933, Fenchel married fellow German refugee mathematician
Käte Sperling.
When
Germany occupied Denmark, Fenchel and roughly eight-thousand other Danish Jews
received refuge in Sweden, where he taught (between 1943 and 1945) at the Danish School in
Lund
Lund (, , ) is a city in the southern Swedish provinces of Sweden, province of Scania, across the Øresund, Öresund strait from Copenhagen. The town had 91,940 inhabitants out of a municipal total of 121,510 . It is the seat of Lund Municipali ...
.
After the Allied powers'
liberation of Denmark
At the outset of World War II in September 1939, Denmark declared itself neutral. For most of the war, the country was a protectorate and then an occupied territory of Germany. The decision to occupy Denmark was taken in Berlin on 17 December ...
, Fenchel returned to Copenhagen.
Professorship postwar
In 1946, Fenchel was elected a member of the
Royal Danish Academy of Sciences and Letters
{{Infobox organization
, name = The Royal Danish Academy of Sciences and Letters
, full_name =
, native_name = Det Kongelige Danske Videnskabernes Selskab
, native_name_lang =
, logo = Royal ...
.
On leave between 1949 and 1951, Fenchel taught in the U.S. at the
University of Southern California
The University of Southern California (USC, SC, or Southern Cal) is a Private university, private research university in Los Angeles, California, United States. Founded in 1880 by Robert M. Widney, it is the oldest private research university in C ...
,
Stanford University
Stanford University, officially Leland Stanford Junior University, is a private research university in Stanford, California. The campus occupies , among the largest in the United States, and enrolls over 17,000 students. Stanford is consider ...
, and
Princeton University
Princeton University is a private university, private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial Colleges, fourth-oldest ins ...
.
From 1952 to 1956 Fenchel was the professor in mechanics at the Polytechnic in Copenhagen.
From 1956 to 1974 he was the professor in mathematics at the
University of Copenhagen
The University of Copenhagen ( da, Københavns Universitet, KU) is a prestigious public university, public research university in Copenhagen, Copenhagen, Denmark. Founded in 1479, the University of Copenhagen is the second-oldest university in ...
.
Last years, death, legacy
Professor Fenchel died on 24 January 1988.
Geometric contributions
Convex geometry
Optimization theory
Fenchel lectured on "Convex Sets, Cones, and Functions" at Princeton University in the early 1950s. His lecture notes shaped the field of
convex analysis
Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory.
Convex sets
A subset C \subseteq X of som ...
, according to the monograph ''Convex Analysis'' of
R. T. Rockafellar.
Hyperbolic geometry
Books
*
*
*
*
*
*
*
See also
*
Convex analysis
Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory.
Convex sets
A subset C \subseteq X of som ...
**
Convex cone
**
Convex function
In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of a function, graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigra ...
**
Convex set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex r ...
**
Legendre–Fenchel transformation
In mathematics and mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known as Legendre–Fenchel transformation, Fenchel transformati ...
*
Convex minimization
**
Fenchel's duality theorem In mathematics, Fenchel's duality theorem is a result in the theory of convex functions named after Werner Fenchel.
Let ''ƒ'' be a proper convex function on R''n'' and let ''g'' be a proper concave function on R''n''. Then, if regularity condi ...
*
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
**
Convex geometry
In mathematics, convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space. Convex sets occur naturally in many areas: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of numbe ...
***
Brunn–Minkowski theorem In mathematics, the Brunn–Minkowski theorem (or Brunn–Minkowski inequality) is an inequality relating the volumes (or more generally Lebesgue measures) of compact subsets of Euclidean space. The original version of the Brunn–Minkowski theor ...
**
Differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
***
Fenchel's theorem
In differential geometry, Fenchel's theorem is an inequality on the total absolute curvature of a closed smooth space curve, stating that it is always at least 2\pi. Equivalently, the average curvature is at least 2 \pi/L, where L is the length ...
**
Hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P ...
***
Jakob Nielsen Jacob or Jakob Nielsen may refer to:
* Jacob Nielsen, Count of Halland (died c. 1309), great grandson of Valdemar II of Denmark
* , Norway (1768-1822)
* Jakob Nielsen (mathematician) (1890–1959), Danish mathematician known for work on automorphi ...
***
Fenchel–Nielsen coordinates In mathematics, Fenchel–Nielsen coordinates are coordinates for Teichmüller space introduced by Werner Fenchel and Jakob Nielsen.
Definition
Suppose that ''S'' is a compact Riemann surface of genus
Genus ( plural genera ) is a taxonomic ...
*
Nonlinear programming
In mathematics, nonlinear programming (NLP) is the process of solving an optimization problem where some of the constraints or the objective function are nonlinear. An optimization problem is one of calculation of the extrema (maxima, minima or s ...
References
External links
*
Werner Fenchel website– contains CV, biography, links to archive, etc.
{{DEFAULTSORT:Fenchel, Werner
Geometers
Differential geometers
20th-century German mathematicians
Danish mathematicians
Jewish emigrants from Nazi Germany to Denmark
1905 births
1988 deaths
Variational analysts