Hugo Hadwiger

Hugo Hadwiger (23 December 1908 in Karlsruhe, Germany – 29 October 1981 in Bern, Switzerland) was a Swiss mathematician, known for his work in geometry, combinatorics, and cryptography.

Although born in Karlsruhe, Germany, Hadwiger grew up in Bern, Switzerland.

He did his undergraduate studies at the University of Bern, where he majored in mathematics but also studied physics and actuarial science.

He continued at Bern for his graduate studies, and received his Ph.D. in 1936 under the supervision of Willy Scherrer.

He was for more than forty years a professor of mathematics at Bern.

Hadwiger's theorem in integral geometry classifies the isometry-invariant valuations on compact convex sets in d-dimensional Euclidean space.

According to this theorem, any such valuation can be expressed as a linear combination of the intrinsic volumes; for instance, in two dimensions, the intrinsic volumes are the area, the perimeter, and the Euler characteristic.

The Hadwiger–Finsler inequality, proven by Hadwiger with Paul Finsler, is an inequality relating the side lengths and area of any triangle in the Euclidean plane.

It generalizes Weitzenböck's inequality and was generalized in turn by Pedoe's inequality.

In the same 1937 paper in which Hadwiger and Finsler published this inequality, they also published the Finsler–Hadwiger theorem on a square derived from two other squares that share a vertex.

Hadwiger's name is also associated with several important unsolved problems in mathematics:

Hadwiger proved a theorem characterizing eutactic stars, systems of points in Euclidean space formed by orthogonal projection of higher-dimensional cross polytopes.

He found a higher-dimensional generalization of the space-filling Hill tetrahedra.

The system was used by the Swiss army and air force between 1947 and 1992.

Asteroid 2151 Hadwiger, discovered in 1977 by Paul Wild, is named after Hadwiger.

And his 1957 book Vorlesungen über Inhalt, Oberfläche und Isoperimetrie was foundational for the theory of Minkowski functionals, used in mathematical morphology.

Hadwiger was one of the principal developers of a Swiss rotor machine for encrypting military communications, known as NEMA.

The Swiss, fearing that the Germans and Allies could read messages transmitted on their Enigma cipher machines, enhanced the system by using ten rotors instead of five.

The first article in the "Research Problems" section of the American Mathematical Monthly was dedicated by Victor Klee to Hadwiger, on the occasion of his 60th birthday, in honor of Hadwiger's work editing a column on unsolved problems in the journal Elemente der Mathematik.