Blaschke selection theorem - Alchetron, the free social encyclopedia

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 a 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

Blaschke selection theorem Wikipedia

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Contents

Alternate statements

Application

References