Blaschke selection theorem

The Blaschke selection theorem is a result in topology 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 Moser worm problem can be shown to have a solution. [cite book|author=Paul J. Kelly|coauthors=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 convex universal cover of minimal size for the collection of planar curves of unit length.

Notes

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