Hadwiger's theorem (convex geometry)

The set of Hadwiger is a tenet of the mathematical area of convex geometry and as such located between the fields of geometry and analysis . It comes from the specialist publication Old and New on convex bodies published by Hugo Hadwiger in 1955 and deals with the polyhedral approximation of certain subsets of Euclidean space by convex polyhedra .

Formulation of the sentence

The sentence can be formulated in the modern version as follows:

For every compact convex neighborhood of zero and each , there is always a compact convex polyhedron with ${\displaystyle N\subseteq \mathbb {R} ^{n}\;(n\in \mathbb {N} )}$ ${\displaystyle \epsilon >0}$ ${\displaystyle P\subseteq \mathbb {R} ^{n}}$

${\displaystyle P\subseteq N\subseteq (1+\epsilon )P}$.

Explanations and Notes

• Hugo Hadwiger only formulated his theorem for egg bodies , i.e. for convex and compact point sets of three-dimensional Euclidean space. He describes a convex polyhedron of three-dimensional Euclidean space as an egg polyhedron .
• A null neighborhood is a set of points in a topological vector space that is the neighborhood of the null vector there .
• For a subset and a real number consists exactly of all with . Is and a convex polyhedron, so called Hadwiger in this context, by dilation with from emanating homothetic polyhedra . ${\displaystyle T\subseteq \mathbb {R} ^{n}\;(n\in \mathbb {N} )}$ ${\displaystyle \lambda }$ ${\displaystyle \lambda T\subseteq \mathbb {R} ^{n}}$ ${\displaystyle \lambda t}$ ${\displaystyle t\in T}$ ${\displaystyle \lambda >1}$ ${\displaystyle P\subseteq \mathbb {R} ^{3}}$ ${\displaystyle \lambda P}$ ${\displaystyle \lambda }$ ${\displaystyle P}$

literature

• Hugo Hadwiger: Old and New about Convex Bodies (=  ELEMENTS OF MATHEMATICS FROM A HIGHER VIEWPOINT . Volume III ). Birkhäuser Verlag , Basel, Stuttgart 1955 ( MR0073220 ).
• Jürg T. Marti : Convex Analysis (=  textbooks and monographs from the field of exact sciences, mathematical series . Volume 54 ). Birkhäuser Verlag , Basel, Stuttgart 1977, ISBN 3-7643-0839-7 ( MR0511737 ).

Individual evidence

1. ↑ Jürg T. Marti: Convex Analysis. 1977, pp. 230-233
2. ↑ Marti, op.cit., P. 231
3. ↑ Hugo Hadwiger: Old and new about convex bodies. 1955, pp. 23-24
4. ↑ Hadwiger, op.cit., P. 8