# Blaschke selection theorem

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

## 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:

## 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.