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 of convex sets contained in a bounded set, the theorem guarantees the existence of a subsequence and a convex set such that converges to 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.{{cite book|author=Paul J. Kelly|author2=Max L. Weiss|title=Geometry and Convexity: A Study in Mathematical Methods|publisher=Wiley|year=1979|pages=Section 6.4}} 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.{{cite journal | last=Wetzel | first=John E. |date=July 2005 | title=The Classical Worm Problem --- A Status Report | journal=Geombinatorics | volume=15 | issue=1 | pages=34–42}}
Notes
References
- {{SpringerEOM|title=Blaschke selection theorem|author=A. B. Ivanov}}
- {{SpringerEOM|title=Metric space of convex sets|author=V. A. Zalgaller}}
- {{cite book|author=Kai-Seng Chou|author2=Xi-Ping Zhu|title=The Curve Shortening Problem|year=2001|publisher=CRC Press|isbn=1-58488-213-1|pages=45}}