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 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.