Werner Fenchel
Encyclopedia
Moritz Werner Fenchel was a mathematician
known for his contributions to geometry
and to optimization theory
. Fenchel established the basic results of convex analysis
and nonlinear optimization theory. Fenchel's monographs and lecture-notes were very influential also. Fenchel lived most of his life in Denmark.
, according to the monograph Convex Analysis of R. T. Rockafellar.
and wrote his doctorate thesis in geometry
(Über Krümmung und Windung geschlossener Raumkurven) under Ludwig Bieberbach
at the University of Berlin. He then taught at the University of Göttingen, until 1933 when the Nazi discrimination laws
led to mass-firings of Jews
.
Fenchel fled to Denmark
in 1933, and then obtained a position at the University of Copenhagen
. When Germany occupied Denmark
, Fenchel and roughly eight-thousand other Danish Jews received refuge
in Sweden. After the liberation of Denmark, Fenchel returned to Copenhagen
.
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
known for his contributions to geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
and to optimization theory
Optimization (mathematics)
In mathematics, computational science, or management science, mathematical optimization refers to the selection of a best element from some set of available alternatives....
. Fenchel established the basic results of convex analysis
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....
and nonlinear optimization theory. Fenchel's monographs and lecture-notes were very influential also. Fenchel lived most of his life in Denmark.
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 analysisConvex 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....
, according to the monograph Convex Analysis of R. T. Rockafellar.
Biographical history
Fenchel was born in GermanyGermany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
and wrote his doctorate thesis in geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
(Über Krümmung und Windung geschlossener Raumkurven) under Ludwig Bieberbach
Ludwig Bieberbach
Ludwig Georg Elias Moses Bieberbach was a German mathematician.-Biography:Born in Goddelau, near Darmstadt, he studied at Heidelberg and under Felix Klein at Göttingen, receiving his doctorate in 1910. His dissertation was titled On the theory of automorphic functions...
at the University of Berlin. He then taught at the University of Göttingen, until 1933 when the Nazi discrimination laws
Law for the Restoration of the Professional Civil Service
The Law for the Restoration of the Professional Civil Service , also known as Civil Service Law, Civil Service Restoration Act, and Law to Re-establish the Civil Service, was a law passed by the National Socialist regime on April 7, 1933, two months after Adolf...
led to mass-firings of Jews
Racial policy of Nazi Germany
The racial policy of Nazi Germany was a set of policies and laws implemented by Nazi Germany, asserting the superiority of the "Aryan race", and based on a specific racist doctrine which claimed scientific legitimacy...
.
Fenchel fled to Denmark
Denmark
Denmark is a Scandinavian country in Northern Europe. The countries of Denmark and Greenland, as well as the Faroe Islands, constitute the Kingdom of Denmark . It is the southernmost of the Nordic countries, southwest of Sweden and south of Norway, and bordered to the south by Germany. Denmark...
in 1933, and then obtained a position at the University of Copenhagen
University of Copenhagen
The University of Copenhagen is the oldest and largest university and research institution in Denmark. Founded in 1479, it has more than 37,000 students, the majority of whom are female , and more than 7,000 employees. The university has several campuses located in and around Copenhagen, with the...
. When Germany occupied Denmark
Occupation of Denmark
Nazi Germany's occupation of Denmark began with Operation Weserübung on 9 April 1940, and lasted until German forces withdrew at the end of World War II following their surrender to the Allies on 5 May 1945. Contrary to the situation in other countries under German occupation, most Danish...
, Fenchel and roughly eight-thousand other Danish Jews received refuge
Rescue of the Danish Jews
The rescue of the Danish Jews occurred during Nazi Germany's occupation of Denmark during World War II. On October 1st 1943 Nazi leader Adolf Hitler ordered Danish Jews to be arrested and deported...
in Sweden. After the liberation of Denmark, Fenchel returned to Copenhagen
Copenhagen
Copenhagen is the capital and largest city of Denmark, with an urban population of 1,199,224 and a metropolitan population of 1,930,260 . With the completion of the transnational Øresund Bridge in 2000, Copenhagen has become the centre of the increasingly integrating Øresund Region...
.
See also
- Convex analysisConvex analysisConvex 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 coneConvex coneIn linear algebra, a convex cone is a subset of a vector space over an ordered field that is closed under linear combinations with positive coefficients.-Definition:...
- Convex functionConvex functionIn mathematics, a real-valued function f defined on an interval is called convex if the graph of the function lies below the line segment joining any two points of the graph. Equivalently, a function is convex if its epigraph is a convex set...
- Convex setConvex setIn Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...
- Legendre–Fenchel transformation
- Convex cone
- Convex minimization
- Fenchel's duality theoremFenchel's duality theoremIn 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 Rn and let g be a proper concave function on Rn...
- Fenchel's duality theorem
- GeometryGeometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
- Convex geometryConvex geometryConvex geometry is the branch of geometry studying convex sets, mainly in Euclidean space.Convex sets occur naturally in many areas of mathematics: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of numbers, integral geometry, linear programming,...
- Brunn-Minkowski theoremBrunn-Minkowski theoremIn mathematics, the Brunn–Minkowski theorem is an inequality relating the volumes of compact subsets of Euclidean space. The original version of the Brunn–Minkowski theorem applied to convex sets; the generalization to compact nonconvex sets stated here is due to L.A...
- Brunn-Minkowski theorem
- Differential geometry
- Hyperbolic geometryHyperbolic geometryIn mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
- Jakob NielsenJakob Nielsen (mathematician)Jakob Nielsen was a Danish mathematician known for his work on automorphisms of surfaces. He was born in the village Mjels on the island of Als in North Schleswig, in modern day Denmark. His mother died when he was 3, and in 1900 he went to live with his aunt and was enrolled in the Realgymnasium...
- Fenchel–Nielsen coordinatesFenchel–Nielsen coordinatesIn mathematics, Fenchel–Nielsen coordinates are coordinates for Teichmüller space introduced by Werner Fenchel and Jakob Nielsen.-Definition:...
- Jakob Nielsen
- Convex geometry
- Nonlinear optimization
External links
- Werner Fenchel website – contains CV, biography, links to archive, etc.