Blaschke selection theorem
The Blaschke selection theorem is a result in topology and convex geometry about sequences of convex sets. Specifically, given a sequence of convex sets contained in a bounded set, the theorem guarantees the existence of a subsequence and a convex set such that converges to in the Hausdorff metric. The theorem is named for Wilhelm Blaschke.
- 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).
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.
- Paul J. Kelly; Max L. Weiss (1979). Geometry and Convexity: A Study in Mathematical Methods. Wiley. pp. Section 6.4.
- Wetzel, John E. (July 2005). "The Classical Worm Problem --- A Status Report". Geombinatorics. 15 (1): 34–42.
- A. B. Ivanov (2001) , "Blaschke selection theorem", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
- V. A. Zalgaller (2001) , "Metric space of convex sets", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
- Kai-Seng Chou; Xi-Ping Zhu (2001). The Curve Shortening Problem. CRC Press. p. 45. ISBN 1-58488-213-1.