|
|
| Line 1: |
Line 1: |
| The '''Blaschke selection theorem''' is a result in [[topology]] and [[convex geometry]] about [[sequence]]s of [[convex set]]s. Specifically, given a sequence <math>\{K_n\}</math> of convex sets contained in a bounded set, the theorem guarantees the existence of a subsequence <math>\{K_{n_m}\}</math> and a convex set <math>K</math> such that <math>K_{n_m}</math> converges to <math>K</math> in the [[Hausdorff metric]]. The theorem is named for [[Wilhelm Blaschke]].
| | Hi there. Allow me begin by introducing the author, her name is Sophia Boon but she by no means really favored that name. [http://www.octionx.sinfauganda.co.ug/node/22469 tarot card readings] Invoicing is what I do for a residing but I've always wanted my own company. The preferred pastime for him and his children is to play lacross and he'll be starting something else along with it. I've always loved living in Alaska.<br><br>Take a look at my blog post - [http://breenq.com/index.php?do=/profile-1144/info/ real psychic readings] readings - [http://Netwk.Hannam.ac.kr/xe/data_2/85669 Netwk.Hannam.ac.kr], |
| | |
| ==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 inequality|isoperimetric problem]] can be shown to have a solution.<ref name=KellyWeiss>{{cite book|author=Paul J. Kelly|coauthors=Max L. Weiss|title=Geometry and Convexity: A Study in Mathematical Methods|publisher=Wiley|year=1979|pages=Section 6.4}}</ref> 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,<ref name=KellyWeiss/>
| |
| * the maximum inclusion problem,<ref name=KellyWeiss/>
| |
| * and the [[Moser's worm problem]] for a convex universal cover of minimal size for the collection of planar curves of unit length.<ref>{{cite journal | last=Wetzel | first=John E. |date=July 2005 | title=The Classical Worm Problem --- A Status Report | journal=Geombinatorics | volume=15 | issue=1 | pages=34–42}}</ref>
| |
| | |
| ==Notes==
| |
| <references/>
| |
| | |
| ==References==
| |
| | |
| * {{SpringerEOM|title=Blaschke selection theorem|urlname=b/b016640|author=A. B. Ivanov}}
| |
| * {{SpringerEOM|title=Metric space of convex sets|urlname=c/c026360|author=V. A. Zalgaller}}
| |
| * {{cite book|author=Kai-Seng Chou|coauthors=Xi-Ping Zhu|title=The Curve Shortening Problem|year=2001|publisher=CRC Press|isbn=1-58488-213-1|pages=45}}
| |
| | |
| {{topology-stub}}
| |
| | |
| [[Category:Geometric topology]]
| |
| [[Category:Compactness theorems]]
| |
| | |
| [[ru:Теорема выбора Бляшке]]
| |
Hi there. Allow me begin by introducing the author, her name is Sophia Boon but she by no means really favored that name. tarot card readings Invoicing is what I do for a residing but I've always wanted my own company. The preferred pastime for him and his children is to play lacross and he'll be starting something else along with it. I've always loved living in Alaska.
Take a look at my blog post - real psychic readings readings - Netwk.Hannam.ac.kr,