Blaschke selection theorem

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The Blaschke selection theorem is a result in topology 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 and a convex set K such that converges to K in the Hausdorff metric. The theorem is named for Wilhelm Blaschke.

Contents 1 Alternate statements 2 Application 3 Notes 4 References

[edit] 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).

[edit] Application

As an example of its use, the Lebesgue universal cover problem can be shown to have a solution.^[1] That is, there exists a convex universal cover of minimal size for the collection of all sets in the plane of unit diameter.

[edit] Notes

1. ^ Paul J. Kelly; Max L. Weiss (1979). Geometry and Convexity: A Study in Mathematical Methods. Wiley. pp. Section 6.4. 

[edit] References

• A. B. Ivanov (2001), "Blaschke selection theorem", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 
• V. A. Zalgaller (2001), "Metric space of convex sets", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 
• Kai-Seng Chou; Xi-Ping Zhu (2001). The Curve Shortening Problem. CRC Press. pp. p45. ISBN 1-58488-213-1.