Set of will

The set of the will is a theorem , which the German mathematician Friedrich Wille (1935-1992) to the mathematical branch of analysis has contributed. The theorem goes back to a work by Willes from 1972 and deals with a coverage problem for bounded subsets in higher-dimensional Euclidean space . It is closely related to several important theorems of mathematics such as Lebesgue's plaster theorem or Borsuk's antipodal theorem . With its help, solvability criteria for nonlinear systems of equations with certain convexity properties can be derived.

Formulation of the sentence

Following the monograph by Jürg T. Marti , the sentence can be stated as follows:

Let there be a finite number of non-empty subsets . The subset is bounded and the other subsets are closed and convex . ${\displaystyle \mathbb {R} ^{n}\;(n\in \mathbb {N} ,n\geq 2)}$ ${\displaystyle T,T_{1},\ldots ,T_{m}\subset \mathbb {R} ^{n}\;(m\in \mathbb {N} )}$ ${\displaystyle T}$ ${\displaystyle T_{j}}$

The subsets should completely cover the - boundary point set , but at the same time points should still lie in the difference set . ${\displaystyle T_{j}}$ ${\displaystyle T}$ ${\displaystyle \partial {T}={\overline {T}}\setminus T^{\circ }}$ ${\displaystyle D=T\setminus \left(\bigcup _{j=1,\ldots ,m}{T_{j}}\right)}$

Then:

(i) . ${\displaystyle m\geq n+1}$

(ii) In the intersection of the subsets is not a single point: . ${\displaystyle m}$ ${\displaystyle T_{j}}$ ${\displaystyle \bigcap _{j=1,\ldots ,m}{T_{j}}=\emptyset }$

(iii) Among the subsets there is a - articulated set sequence , the intersection of which is not empty and which contains a point that is at the same time a contact point of the difference set . ${\displaystyle m}$ ${\displaystyle T_{j}}$ ${\displaystyle n}$ ${\displaystyle T_{j_{1}},\ldots ,T_{j_{n}}}$ ${\displaystyle \bigcap _{k=1,\ldots ,n}{T_{j_{k}}}}$ ${\displaystyle p}$ ${\displaystyle D}$

Corollary

Because of (i)  ! The theorem of Wille entails a corollary which can be stated as follows:

If in -dimensional Euclidean space closed and convex subsets cover the boundary point set of a given restricted subset , then these subsets already cover the entire subset . ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle \partial {T}}$ ${\displaystyle T}$ ${\displaystyle n}$ ${\displaystyle T}$

Related Result: A Set of Mountains

In 1959, the French mathematician Claude Berge (1926–2002) provided a related theorem, which addresses the question under which conditions finitely many closed convex subsets in Euclidean space (and more generally in a given topological vector space ) not another given convex subset cover. Following the monograph by Josef Stoer and Christoph Witzgall, this sentence can be presented as follows:

A topological vector space is given or it is even . ${\displaystyle X}$ ${\displaystyle X=\mathbb {R} ^{n}\;(n\in \mathbb {N} )}$

Furthermore, let there be a finite number of convex subsets , all of which should be closed in . ${\displaystyle T,T_{1},\ldots ,T_{m}\subseteq X\;(m\in \mathbb {N} )}$ ${\displaystyle T_{j}}$ ${\displaystyle X}$

In addition, the following conditions should be met:

(a) For always be ${\displaystyle k=1,\ldots ,m}$

${\displaystyle T\cap \bigcap _{j=1,\ldots ,m\; \atop \;\ j\neq k}{T_{j}}\neq \emptyset }$.

(b) Overall, let

${\displaystyle T\cap \bigcap _{j=1,\ldots ,m}{T_{j}}=\emptyset }$.

Then:

${\displaystyle T\nsubseteq \bigcup _{j=1,\ldots ,m}{T_{j}}}$.

literature

• Claude Berge: Sur une propriété combinatoire des ensembles convexes . In: Comptes rendus de l'Académie des sciences Paris . tape 248 , 1959, pp. 2698-2699 ( MR0106435 ).
• 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 ).
• Josef Stoer , Christoph Witzgall : Convexity and Optimization in Finite Dimensions. I. (=  The basic teachings of the mathematical sciences in individual representations . Volume 163 ). Springer Verlag , Berlin, Heidelberg, New York 1970 ( MR0286498 ).
• Friedrich Wille: Coverings with convex sets and nonlinear systems of equations . In: Commentarii Mathematici Helvetici . tape 47 , 1972, p. 273-288 ( MR0317183 ).

Individual evidence

1. ↑ Jürg T. Marti: Convex Analysis. 1977, p. 214 ff, p. 273
2. ↑ Friedrich Wille: Coverings with convex sets and nonlinear systems of equations. Comment. Math. Helv. 47, pp. 273-288
3. ↑ Marti, op.cit., P. 217
4. ↑ Marti, op.cit., P. 218
5. ^ Josef Stoer, Christoph Witzgall: Convexity and Optimization in Finite Dimensions. I. 1970, pp. 119-121