Blaschke selection theorem

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

Application

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:

Notes

  1. 1 2 3 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


This article is issued from Wikipedia - version of the Friday, March 25, 2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.