Blaschke selection theorem

Short description: Any sequence of convex sets contained in a bounded set has a convergent subsequence

The Blaschke selection theorem is a result in topology and convex geometry about sequences of convex sets. Specifically, given a sequence [math]\displaystyle{ \{K_n\} }[/math] of convex sets contained in a bounded set, the theorem guarantees the existence of a subsequence [math]\displaystyle{ \{K_{n_m}\} }[/math] and a convex set [math]\displaystyle{ K }[/math] such that [math]\displaystyle{ K_{n_m} }[/math] converges to [math]\displaystyle{ K }[/math] in the Hausdorff metric. The theorem is named for Wilhelm Blaschke.

Alternate statements

• A succinct statement of the theorem is that the metric space of convex bodies is locally compact.
• Using the Hausdorff metric on sets, every infinite collection of compact subsets of the unit ball has a limit point (and that limit point is itself a compact set).

Application

As an example of its use, the isoperimetric problem can be shown to have a solution. That is, there exists a curve of fixed length that encloses the maximum area possible. Other problems likewise can be shown to have a solution:

• Lebesgue's universal covering problem for a convex universal cover of minimal size for the collection of all sets in the plane of unit diameter,
• the maximum inclusion problem,
• and the Moser's worm problem for a convex universal cover of minimal size for the collection of planar curves of unit length.

References

• Hazewinkel, Michiel, ed. (2001), "Blaschke selection theorem", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page 
• Hazewinkel, Michiel, ed. (2001), "Metric space of convex sets", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page 
• Kai-Seng Chou; Xi-Ping Zhu (2001). The Curve Shortening Problem. CRC Press. pp. 45. ISBN 1-58488-213-1. 

ru:Теорема выбора Бляшке