Hugo Hadwiger (23 December 1908 inKarlsruhe, Germany – 29 October 1981 inBern, Switzerland)^[1] was aSwissmathematician, known for his work ingeometry,combinatorics, andcryptography.

Although born inKarlsruhe, Germany, Hadwiger grew up inBern, Switzerland.^[2] He did his undergraduate studies at theUniversity of Bern, where he majored in mathematics but also studied physics andactuarial science.^[2] He continued at Bern for his graduate studies, and received his Ph.D. in 1936 under the supervision of Willy Scherrer.^[3] He was for more than forty years a professor of mathematics at Bern.^[4]

Hadwiger's theorem inintegral geometry classifies the isometry-invariantvaluations oncompactconvex sets ind-dimensional Euclidean space. According to this theorem, any such valuation can be expressed as a linear combination of theintrinsic volumes; for instance, in two dimensions, the intrinsic volumes are thearea, theperimeter, and theEuler characteristic.^[5]

TheHadwiger–Finsler inequality, proven by Hadwiger withPaul Finsler, is an inequality relating the side lengths and area of anytriangle in theEuclidean plane.^[6] It generalizesWeitzenböck's inequality and was generalized in turn byPedoe's inequality. In the same 1937 paper in which Hadwiger and Finsler published this inequality, they also published theFinsler–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:

• TheHadwiger conjecture in graph theory, posed by Hadwiger in 1943^[7] and called byBollobás, Catlin & Erdős (1980) “one of the deepest unsolved problems in graph theory,”^[8] describes a conjectured connection betweengraph coloring andgraph minors. TheHadwiger number of a graph is the number of vertices in the largestclique that can be formed as a minor in the graph; the Hadwiger conjecture states that this is always at least as large as thechromatic number.
• TheHadwiger conjecture in combinatorial geometry concerns the minimum number of smaller copies of a convex body needed to cover the body, or equivalently the minimum number of light sources needed to illuminate the surface of the body; for instance, in three dimensions, it is known that any convex body can be illuminated by 16 light sources, but Hadwiger's conjecture implies that only eight light sources are always sufficient.^[9][10]
• TheHadwiger–Kneser–Poulsen conjecture states that, if the centers of a system of balls in Euclidean space are moved closer together, then the volume of the union of the balls cannot increase. It has been proven in the plane, but remains open in higher dimensions.^[11]
• TheHadwiger–Nelson problem concerns the minimum number of colors needed to color the points of the Euclidean plane so that no two points at unit distance from each other are given the same color. It was first proposed byEdward Nelson in 1950. Hadwiger popularized it by including it in a problem collection in 1961;^[12][13] already in 1945 he had published a related result, showing that any cover of the plane by five congruent closed sets contains a unit distance in one of the sets.^[14]

Hadwiger proved a theorem characterizingeutactic stars, systems of points in Euclidean space formed byorthogonal projection of higher-dimensionalcross polytopes. He found a higher-dimensional generalization of the space-fillingHill tetrahedra.^[15] And his 1957 bookVorlesungen über Inhalt, Oberfläche und Isoperimetrie was foundational for the theory ofMinkowski functionals, used inmathematical morphology.^{[citation needed]}

Hadwiger was one of the principal developers of a Swissrotor machine for encrypting military communications, known asNEMA. The Swiss, fearing that the Germans and Allies could read messages transmitted on theirEnigma cipher machines, enhanced the system by using ten rotors instead of five. The system was used by the Swiss army and air force between 1947 and 1992.^[16]

Asteroid2151 Hadwiger, discovered in 1977 byPaul Wild, is named after Hadwiger.^[4]

The first article in the "Research Problems" section of theAmerican Mathematical Monthly was dedicated byVictor Klee to Hadwiger, on the occasion of his 60th birthday, in honor of Hadwiger's work editing a column on unsolved problems in the journalElemente der Mathematik.^[2]

• Altes und Neues über konvexe Körper, Birkhäuser 1955^[17]
• Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer, Grundlehren der mathematischen Wissenschaften, 1957^[18]
• with H. Debrunner, V. KleeCombinatorial Geometry in the Plane, Holt, Rinehart and Winston, New York 1964;Dover reprint 2015

• "Über eine Klassifikation der Streckenkomplexe", Vierteljahresschrift der Naturforschenden Gesellschaft Zürich, vol. 88, 1943, pp. 133–143 (Hadwiger's conjecture in graph theory)
• with Paul GlurZerlegungsgleichheit ebener Polygone, Elemente der Math, vol. 6, 1951, pp. 97-106
• Ergänzungsgleichheit k-dimensionaler Polyeder, Math. Zeitschrift, vol. 55, 1952, pp. 292-298
• Lineare additive Polyederfunktionale und Zerlegungsgleichheit, Math. Z., vol. 58, 1953, pp. 4-14
• Zum Problem der Zerlegungsgleichheit k-dimensionaler Polyeder, Mathematische Annalen vol. 127, 1954, pp. 170–174

• 1908 births
• 1981 deaths
• Modern cryptographers
• 20th-century Swiss mathematicians
• Scientists from Karlsruhe
• Scientists from Bern
• University of Bern alumni
• Combinatorialists
• Geometers
• German emigrants to Switzerland

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