Minkowski's theorem

The set of Minkowski (after Hermann Minkowski ) is a mathematical theorem which is concerned with certain geometric shapes and their outermost edge points. More precisely, it comes from the theory of convex sets in finite-dimensional spaces and establishes a relationship between a compact convex set and its extreme points .

Formulation of the sentence

For a compact, convex set and a subset , the following statements are equivalent: ${\displaystyle C\subset \mathbb {R} ^{d}}$ ${\displaystyle M\subset C}$

• ${\displaystyle C}$is the convex hull of . ${\displaystyle M}$
• The extremal points of are contained in. ${\displaystyle C}$ ${\displaystyle M}$

In particular, in a finite-dimensional space a compact, convex set is equal to the convex hull of its extremal points. This statement is also often called Minkowski's theorem .

Theorem of Carathéodory

The mathematician Constantin Carathéodory proved the following well-known theorem in 1911 :

(1) If (for two given natural numbers and with ) in Euclidean space , a subset added, and this in an n-dimensional affine subspace of included so is the convex hull of the same as the set of all convex combinations consisting of a maximum of elements of formed become. Expressed formally, the following applies: ${\displaystyle n}$ ${\displaystyle d}$ ${\displaystyle n\leq d}$ ${\displaystyle \mathbb {R} ^{d}}$ ${\displaystyle M\subset \mathbb {R} ^{d}}$ ${\displaystyle \mathbb {R} ^{d}}$ ${\displaystyle M}$ ${\displaystyle n+1}$ ${\displaystyle M}$

${\displaystyle \operatorname {conv} {M}=\bigcup _{T\subseteq M\;,\;|T|\leq n+1}{\operatorname {conv} {T}}}$.

If you combine this with Minkowski's theorem, you get:

(2) Every point of a compact , convex subset that is contained in an n-dimensional affine subspace is a convex combination of at most extreme points. ${\displaystyle C\subset \mathbb {R} ^{d}}$ ${\displaystyle n+1}$

Since you can always choose an affine subspace, you get a statement that is sometimes also referred to as Minkowski's theorem : ${\displaystyle \mathbb {R} ^{d}}$

(3) Every point of a compact, convex subset is a convex combination of at most extreme points. ${\displaystyle C\subset \mathbb {R} ^{d}}$ ${\displaystyle d+1}$

Generalization of Carathéodory's theorem

In 1982, the set Hungarian mathematician Imre Bárány before a generalization of the Carathéodory'schen set, the one as set of Bárány ( English Bárány's Theorem can) and denote the following states:

(4) If subsets are given as well as a point in space , there are always selected points in space in such a way that these points in space already lie in the convex envelope . ${\displaystyle d+1}$ ${\displaystyle T_{1},\ldots ,T_{d+1}\subseteq \mathbb {R} ^{d}}$ ${\displaystyle x_{0}\in \left(\bigcap _{j=1,\ldots ,d+1}{\operatorname {conv} {T_{j}}}\right)}$ ${\displaystyle d+1}$ ${\displaystyle x_{j}\in T_{j}\;(j=1,\ldots ,d+1)}$ ${\displaystyle x_{0}}$ ${\displaystyle \operatorname {conv} {\{x_{1},\ldots ,x_{d+1}\}}}$ ${\displaystyle d+1}$

One obtains Carathéodory's theorem for the special case . ${\displaystyle T_{1}=T_{2}=\cdots =T_{d+1}}$

Remarks

• Minkowski's theorem above generalizes in infinite-dimensional locally convex spaces to the Kerin-Milman theorem . The statements that apply there are weaker, as qualifications are added.
• The above statement (3) cannot be improved any further. To represent the center of a non-degenerate simplex in all you have to use corners. ${\displaystyle \mathbb {R} ^{d}}$ ${\displaystyle d+1}$
• Another non-trivial conclusion from Minkowski's theorem is that a compact, convex set has extremal points at all. Such considerations play a role in the justification of the simplex method .

literature

• WA Coppel: Foundations of Convex Geometry (=  Australian Mathematical Society Lecture Series . Volume 12 ). Cambridge University Press , Cambridge 1998, ISBN 0-521-63970-0 ( MR1629043 ).
• Steven R. Lay: Convex Sets and Their Applications . John Wiley & Sons , New York, Chichester, Brisbane, Toronto, Singapore 1982, ISBN 0-471-09584-2 .
• Kurt Leichtweiß : Convex quantities (=  university text ). Springer Verlag , Berlin, Heidelberg, New York 1980, ISBN 3-540-09071-1 ( MR0586235 ).
• 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. ↑ Arne Brøndsted: An Introduction to Convex Polytopes , Springer New York Heidelberg Berlin (1983), Th.5.10
2. ↑ C. Carathéodory: About the range of variability of Fourier's constants of positive harmonic functions . In: Rendiconti del Circolo Matematico di Palermo . tape 32 , 1911, pp. 193-217 .
3. ^ Arne Brøndsted: An Introduction to Convex Polytopes , Springer New York Heidelberg Berlin (1983), Cor. 2.4
4. ^ WA Coppel: Foundations of Convex Geometry. 1998, p. 67
5. ↑ Imre Bárány: A generalization of Carathéodory's theorem . In: Discrete Mathematics . tape 40 , 1982, pp. 141-152 ( MR0676720 ).
6. ↑ ^{a b} Coppel, op.cit., P. 68