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Besicovitch Covering Lemma

  1. Dec 5, 2013 #1
    1. The problem statement, all variables and given/known data

    Let {Q} be a collection of cubes covering a set E in R^n. Prove that there is a countable sub collection {Q}' of these cubes which covers E and [itex] \cup (\frac{1}{2}){Q} \subseteq \cup {Q}' [/itex], and the number of cubes in the subcollection containing any given point of E is less than something depending only on the dimension.
    Oh and the sup of the side lengths of the cubes is bounded, but E is not bounded.

    2. Relevant equations

    [itex]\frac{1}{2}Q [/itex] means the cube inside Q with side length 1/2 of the side length of Q.
    In other words, the countable subcovering needs to cover the middle sections of the original cover.

    3. The attempt at a solution

    I really like this proof http://books.google.com/books?id=5d...=onepage&q=besicovitch covering cubes&f=false

    But in this proof the set E is the centers of the cubes, so we only end up covering centers, not middle halves, and also E is bounded in that proof. My professor said I could modify this proof but I don't see how. *sad face* He also said the first half of my most recent attempt at this proof was badly written and the second half was worse. :(:(

    The idea for dealing with the unbounded set E is to take "layers" of cubes which are about the same size. So something like sup{S(Q)}=R where S(Q) is the side length of Q and then [itex](1-\epsilon)^kR< S(q)\le R[/itex] for the first layer, then take a maximal set of points in E which are some specific distance apart (this is a piece I messed up on, no matter what distance I separate my points by there is always some flaw in my choice). Then take the cubes containing those points and discard some cubes (which ones?). Rinse and repeat.

    I tried doing some optimal arrangement of separated points around a specific point for the bound, but I was told I did that wrong too :( (that was the "worse" part apparently.)
     
  2. jcsd
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