Blaschke's selection sentence

The selection set of Blaschke ( engl. Blaschke Selection theorem ) is a mathematical theorem which a convergence problem of Convex treated. The sentence is to be assigned to the transition field between convex geometry and topology . It was presented by the geometer Wilhelm Blaschke in his writing Kreis und Kugel in 1916.

Formulation of the selection sentence

Blaschke's selection sentence can be formulated in the modern version as follows:

Given is a sequence of non-empty compact convex subsets of a finite-dimensional normed vector space over . If these subsets are uniformly bounded in the sense that they are all covered by a compact subset of , a subsequence can be selected which converges in the Hausdorff metric to a non-empty compact convex subset of . ${\displaystyle (C_{n})_{n\in \mathbb {N} }}$ ${\displaystyle {\mathcal {L}}}$ ${\displaystyle {\mathbb {R} }}$ ${\displaystyle {\mathcal {L}}}$ ${\displaystyle (C_{n_{k}})_{k\in \mathbb {N} }}$ ${\displaystyle C}$ ${\displaystyle {\mathcal {L}}}$

Different formulation of the selection sentence

If one denotes with the set system of the non-empty compact convex subsets of the normalized vector space and with the Hausdorff metric , then the selection theorem says: ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {L}}}$ ${\displaystyle h}$ ${\displaystyle {\mathcal {C}}}$

${\displaystyle {\mathcal {(}}{\mathcal {C}},h)}$is a locally compact metric space .

Applications

The selection theorem is often used where proof of existence has to lead to extremal problems of convex geometry. As Wilhelm Blaschke already shows in the circle and sphere , the isoperimetric inequality , for example, can be derived with the help of the selection set .

Related results

Blaschke's selection theorem is a consequence of Arzelà-Ascoli's theorem and, in a generalized version, even proves to be equivalent to that (in the classical form).

literature

• Wilhelm Blaschke : circle and sphere . Chelsea Publishing Company , New York 1949 (reprint of the Leipzig edition, 1916).

• Tommy Bonnesen , Werner Fennel : Theory of convex bodies. Corrected reprint. Springer-Verlag , Berlin et al. 1974, ISBN 3-540-06234-3 .

• Peter M. Gruber : Convex and Discrete Geometry . Springer-Verlag, Berlin et al. 2007, ISBN 978-3-540-71132-2 .

• Hugo Hadwiger : Lectures on content, surface and isoperimetry . Springer-Verlag, Berlin et al. 1957, ISBN 3-540-02151-5 .

• Steven R. Lay: Convex sets and their applications . John Wiley & Sons , New York et al. 1982, ISBN 0-471-09584-2 .

• Jürg T. Marti: Convex Analysis . Birkhäuser Verlag , Basel et al. 1977, ISBN 3-7643-0839-7 .

• Frederick A. Valentine: Convex sets (=  BI university pocket books 402 / 402a ). Bibliographisches Institut , Mannheim 1968, DNB  458498998 .

Individual evidence

1. ^ W. Blaschke: circle and sphere . 1949, p. 62 .
2. ^ PM Gruber: Convex and Discrete Geometry . 2007, p. 85 .
3. ^ SR Lay: Convex sets and their applications . 1982, p. 98 .
4. ^ JT Marti: Convex Analysis . 1977, p. 226 .
5. ^ FA Valentine: Convex sets . 1968, p. 47 .
6. ^ H. Hadwiger: Lectures on content, surface and isoperimetry . 1957, p. 154 .
7. ^ JT Marti: Convex Analysis . 1977, p. 220 .
8. ^ SR Lay: Convex sets and their applications . 1982, p. 101 .
9. ^ W. Blaschke: circle and sphere . 1949, p. 79 ff .
10. ^ PM Gruber: Convex and Discrete Geometry . 2007, p. 84-88 .