Blaschke selection theorem

Template:Short description 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.

References

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