Blaschke selection theorem

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

Alternate statements

• A succinct statement of the theorem is that a 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 universal cover 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

• A. B. Ivanov (2001), "Blaschke selection theorem", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=b/b016640 
• V. A. Zalgaller (2001), "Metric space of convex sets", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=c/c026360 
• Kai-Seng Chou; Xi-Ping Zhu (2001). The Curve Shortening Problem. CRC Press. pp. 45. ISBN 1-58488-213-1.