mathematics

Werner Fenchel

Werner Fenchel is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Werner Fenchel rather than just read about it. In short: Moritz Werner Fenchel (German: [ˈfɛnçəl]; 3 May 1905 – 24 January 1988) was a German-Danish mathematician known for his contributions to geometry and to optimization theory. Fenchel established the basic results of convex analysis and nonlinear optimization theory which would, in time, serve as the foundation for nonlinear programming.

Key takeaways

Werner Fenchel belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Werner Fenchel to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Werner Fenchel from memory before moving on to harder problems.

Reference excerpt

Moritz Werner Fenchel (German: [ˈfɛnçəl]; 3 May 1905 – 24 January 1988) was a German-Danish mathematician known for his contributions to geometry and to optimization theory. Fenchel established the basic results of convex analysis and nonlinear optimization theory which would, in time, serve as the foundation for nonlinear programming. 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, Germany, his younger brother was the Israeli film director and architect Heinz Fenchel. Fenchel studied mathematics and physics at the University of Berlin between 1923 and 1928. He wrote his doctorate thesis in geometry (Über Krümmung und Windung geschlossener Raumkurven) under Ludwig Bieberbach.

Professorship in Germany From 1928 to 1933, Fenchel was Professor E. Landau's Assistant at the University of Göttingen. During a one-year leave (on Rockefeller Fellowship) between 1930 and 1931, Fenchel spent time in Rome with Tullio Levi-Civita, as well as in Copenhagen with Harald Bohr and Tommy Bonnesen. 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. 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. After the Allied powers' liberation of Denmark, Fenchel returned to Copenhagen.

Professorship postwar In 1946, Fenchel was elected a member of the Royal Danish Academy of Sciences and Letters. On leave between 1949 and 1951, Fenchel taught in the U.S. at the University of Southern California, Stanford University, and Princeton University. 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.

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, according to the monograph Convex Analysis of R. T. Rockafellar.

Hyperbolic geometry

Books

Illustrations

Worked examples

Example 1 — a first encounter with Werner Fenchel

Start with the simplest possible case. Write down what Werner Fenchel claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Werner Fenchel before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Werner Fenchel ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Werner Fenchel

In research: Werner Fenchel appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Werner Fenchel in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Werner Fenchel is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1905 births, 1988 deaths, 20th-century German mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Werner Fenchel outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Werner Fenchel in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Werner Fenchel means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Werner Fenchel out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Werner Fenchel in simple terms?

Why does Werner Fenchel matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Werner Fenchel?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Werner Fenchel.