Blaschke selection theorem

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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[edit]

  • 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[edit]

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 universal cover 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[edit]

  1. ^ a b c 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[edit]