Re: math -- finite union of rectangular regions

On Mar 31, 3:23 am, quasi <qu...@xxxxxxxx> wrote:
On Sun, 30 Mar 2008 20:33:41 -0700 (PDT), Chip Eastham



<hardm...@xxxxxxxxx> wrote:
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.

Sounds interesting.

quasi

Let A be a finite set of closed rectangles in the plane
whose sides are parallel to the coordinate axes whose
union is simply connected. Let us assume that

(*) for all R in A, the union of the rectangles
in A - { R } is not simply connected.

For each rectangle R in A, let

C(R) = R - union { R' in A : R' != R }

Let us call each of the connected components of C(R) such
that its closure is contained in the interior of R a *hole*
of A, and let us say that it belongs to R.

A hole of A belongs to exactly one rectangle in R, which is
the only one which contains it. The holes are moreover disjoint.

Conversely, our hypothesis (*) is equivalent to the
statement that to each rectangle in R belongs at least one
hole.

For each bounded subset X of the plane, let us call the sup
of the y coordinates of the points in X the *height* of X.

Let as pick a hole H of minimal height among the
holes of A. The boundary B of H is a finite polygonal
closed arc, whose segments are parallel to the coordinate
axes. Pick one horizontal segment in B which has minimal
height among the horizontal segments of B. This segment
shares a subsegment of positive length with the boundary
of some rectangle S in A. Let H' be one of the holes in S.
It is clear that the height of H' is strictly less that the
weight of H. This is absurd.

Therefore (*) cannot hold.

-- m
.

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