Tietze's theorem (convex geometry)

In convex geometry , one of the branches of mathematics , Tietze's theorem is one of those theorems which deal with the question of characterizing the convexity of subsets of Euclidean space and (more generally) of real linear Hausdorff spaces with the help of local support properties . The proposition is thus located in the transition field between geometry and the theory of topological vector spaces . It is essentially based on a scientific work by the mathematician Heinrich Tietze from 1929.

Formulation of the sentence

The sentence can be summarized as follows:

Is a Hausdorff topological vector space given and a therein open and connected subset , which in each of its edge points supported locally weak is, it is in convex. This is especially true for the case that the - is dimensional Euclidean space . ${\displaystyle \mathbb {R} }$ ${\displaystyle X}$ ${\displaystyle U\subseteq X}$ ${\displaystyle U}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle n}$ ${\displaystyle \mathbb {R} ^{n}\;(n\in \mathbb {N} )}$

Related results

Tietze's theorem was preceded by a theorem which was proven by a number of important mathematicians, not least by Constantin Carathéodory in 1907 and by Hermann Brunn and Hermann Minkowski in 1910. It can be formulated as follows:

Is a Hausdorff topological vector space given and a therein closed subset having at least one interior point , the subset is in then convex exactly when each of its edge points by a supporting hyperplane of passes. ${\displaystyle \mathbb {R} }$ ${\displaystyle X}$ ${\displaystyle A\subseteq X}$ ${\displaystyle x\in A^{\circ }}$ ${\displaystyle A}$ ${\displaystyle X}$ ${\displaystyle A}$

In differential geometry , another theorem is known, presented by Jacques Hadamard in 1897:

An egg surface in three-dimensional Euclidean space is strictly convex in the sense that for every point in space it contains, the surface is located entirely on one side of the tangential plane adjacent to it. ${\displaystyle M}$ ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle p\in M}$ ${\displaystyle M}$ ${\displaystyle p}$ ${\displaystyle T_{p}{M}}$

Explanations

• As usual, Euclidean space is regarded as having the standard scalar product (as well as the geometric and metric structure given with it) and, in particular, as being provided with the Euclidean distance function . ${\displaystyle \mathbb {R} ^{n}\;(n\in \mathbb {N} )}$
• In relation to a (Hausdorff) topological vector space , a measure inherent in such subset and a -Randpunkt they say, will in supported weakly locally if there is an environment of there and a not with the zero functional identical linear functional so that the following applies: From and always follows . ${\displaystyle X}$ ${\displaystyle T\subseteq X}$ ${\displaystyle T}$ ${\displaystyle x\in \partial {T}}$ ${\displaystyle T}$ ${\displaystyle x}$ ${\displaystyle U(x)}$ ${\displaystyle x}$ ${\displaystyle f\colon X\rightarrow \mathbb {R} }$ ${\displaystyle y\in U(x)\setminus \{x\}}$ ${\displaystyle f(y)>f(x)}$ ${\displaystyle y\not \in T}$
• A subset located in three-dimensional Euclidean space is an egg surface if it is a compact regular surface there and has positive Gaussian curvature in each of its points . The term goes back to Wilhelm Blaschke . ${\displaystyle M\subset \mathbb {R} ^{3}}$
• Every tangent plane to a point on a regular surface is a hyperplane of three-dimensional Euclidean space. ${\displaystyle T_{p}{M}}$ ${\displaystyle p}$ ${\displaystyle M}$
• A hyperplane includes the covering of the by the two associated closed half-spaces , which is such that each point in space lies in one of the two. If a given subset is here either a subset of one or a subset of the other, it is said that it is entirely on one side of the hyperplane . ${\displaystyle H\subset \mathbb {R} ^{3}}$ ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle T\subset \mathbb {R} ^{3}}$ ${\displaystyle T}$

literature

• H. Brunn: On the theory of the egg regions . In: Archives of Mathematics and Physics . tape 17 , 1910, pp. 289-300 .
• C. Carathéodory: About the range of variability of the coefficients of power series that do not take on given values . In: Mathematical Annals . tape 64 , 1907, pp. 95-115 ( MR1511425 ).
• J. Hadamard: Sur certaines propriétés des trajectoires en dynamique . In: Journal de mathématiques pures et appliquées (5) . tape 3 , 1897, p. 331-387 .
• Wilhelm Klingenberg : A lecture on differential geometry (=  Heidelberg pocket books . Volume 107 ). Springer Verlag , Berlin, Heidelberg, New York 1973 ( MR0415512 ).
• Steven R. Lay : Convex Sets and Their Applications (=  Pure and Applied Mathematics ). John Wiley & Sons , New York, Chichester, Brisbane, Toronto, Singapore 1982, ISBN 0-471-09584-2 ( MR0655598 ).
• Hermann Minkowski: Geometry of Numbers . BG Teubner Verlag , Leipzig, Berlin: 1910, p. 1-256 ( [1] ).
• Heinrich Tietze: Remarks on convex and non-convex figures . In: Journal for pure and applied mathematics . tape 160 , 1929, pp. 67-69 ( MR1581176 ).
• Frederick A. Valentine : Convex Sets . Translation from English by E. Heil (=  BI university paperbacks . Volume 402 / 402a). Bibliographisches Institut , Mannheim 1968 ( MR0226495 ).

Individual evidence

1. ↑ Frederick A. Valentine: Convex Sets. 1964, pp. 57-66
2. ↑ Steven R. Lay: Convex Sets and Their Applications. 1982, pp. 104-115
3. ↑ Valentine, op.cit., P. 63
4. ↑ Lay, op.cit., P. 110
5. ↑ Valentine, op.cit., P. 57
6. ^ Wilhelm Klingenberg: A lecture on differential geometry. 1973, p. 100
7. ↑ According to the presentation in Klingenberg's A Lecture on Differential Geometry. Hadamard proved even more and in particular that every egg-surface in is an orientable -dimensional manifold . ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle 2}$