Mathematical Concepts Named After Hadwiger
Hadwiger's theorem in integral geometry classifies the possible isotropic measures on compact convex sets in d-dimensional Euclidean space. According to this theorem, any such measure can be expressed as a linear combination of d + 1 fundamental measures; for instance, in two dimensions, there are three possible measures of this type, one corresponding to area, a second corresponding to perimeter, and a third corresponding to 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.
Hadwiger's name is also associated with several important unsolved problems in mathematics:
- The Hadwiger conjecture in graph theory, posed by Hadwiger in 1943 and called by Bollobás, Catlin & Erdős (1980) “one of the deepest unsolved problems in graph theory,” describes a conjectured connection between graph coloring and graph minors. The Hadwiger number of a graph is the number of vertices in the largest clique that can be formed as a minor in the graph; the Hadwiger conjecture states that this is always at least as large as the chromatic number.
- The Hadwiger 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.
- The Hadwiger–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.
- The Hadwiger–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 by E. Nelson in 1950; Hadwiger popularized it by including it in a problem collection in 1961. However, in 1945, Hadwiger had published a closely related result, showing that any cover of the plane by five congruent closed sets contains a unit distance in one of the sets.
Read more about this topic: Hugo Hadwiger
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