# 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

• 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.[1] 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,[1]
• the maximum inclusion problem,[1]
• and the Moser's worm problem for a convex universal cover of minimal size for the collection of planar curves of unit length.[2]

## Notes

1. ^ a b c Paul J. Kelly; Max L. Weiss (1979). Geometry and Convexity: A Study in Mathematical Methods. Wiley. pp. Section 6.4.
2. ^ Wetzel, John E. (July 2005). "The Classical Worm Problem --- A Status Report". Geombinatorics 15 (1): 34–42.