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.

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.^[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's universal covering 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. 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.

References

• A. B. Ivanov (2001) [1994], "Blaschke selection theorem", Encyclopedia of Mathematics, EMS Press
• V. A. Zalgaller (2001) [1994], "Metric space of convex sets", Encyclopedia of Mathematics, EMS Press
• Kai-Seng Chou; Xi-Ping Zhu (2001). The Curve Shortening Problem. CRC Press. p. 45. ISBN 1-58488-213-1.