Re: math -- finite union of rectangular regions
- From: Chip Eastham <hardmath@xxxxxxxxx>
- Date: Sun, 30 Mar 2008 20:33:41 -0700 (PDT)
On Mar 29, 11:31 pm, quasi <qu...@xxxxxxxx> wrote:
Trying to avoid counterexamples, while still saying something
nontrivial ...
Conjecture:
If n closed, nondegenerate, rectangular regions in R^2, where n>1, and
where all rectangle sides are either horizontal or vertical, is such
that the union is contractible, then it is possible to omit one of the
regions from the union, so that the union of the remaining regions is
still contractible.
Remark:
If this conjecture manages to hold up, consider the generalization to
m-dimensional rectangular regions in R^m.
quasi
With rectangles whose sides are parallel to the coordinate
axes, I think we can make some arguments based on minimizing
the sup of x-coordinates in the "hole" uncovered by removing
one rectangle. I need to write it up more carefully, but I
think a proof by contradiction is possible.
regards, chip
.
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