Blaschke selection theorem
# 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.
