Re: Mike Throll 5

In article <3ba...@News.CIS.DFN.DE>, Timofei Shatrov wrote:
>On 17 Sep 2001 17:38:49 GMT, uncle monty <unc...@babyfishmouth.org> tried to
>confuse everyone with this message:
>
>>Given an initial separation of the plane RxR into a subset A and not-A,
>>and an iterative process which at each step assigns membership in A to
>>each point outside A whose circular neighbourhood of radius 1 has a
>>greater intersection with A than with not-A, and removes each point
>>previously within A whose circular neighbourhood of radius 1 has a greater
>>intersection with not-A than with A, is it possible to choose initial A
>>such that its area will increase by a factor of 1000 or more during the
>>process?
>>
>>Tell me, does the blot have to be simply connected in the beginning?
>I think yes. It also has to be finite.

Obviously it must have a well-defined finite area, for the question to
make sense.  Other than that, finiteness makes little difference -- see
below.


>>Do we need the final limiting area to be more than 1000 times the
>>initial area, or can some intermediate area be 1000 times?
>
>I think all finite figures will eventually die. You need to make it 1000 times
>larger only at one turn.

Yes, all finite figures must eventually vanish.  It's easy to see that
all convex figures will vanish, and that if a figure vanishes, all its
sub-figures must also vanish.  As any finite figure may surrounded by a
convex hull, the proof follows.

The use of the word "finite" above is slightly vague, but things are
simplified by the fact that any figure with a finite area must after one
iteration be a subset of a circle with a finite diameter.  Proof by
contradiction left as an exercise for the reader.

This offers no direct help with the original problem, of course.


>>I imagine it is some 
>>impossible fractal shape to begin with.
>
>No way. It's not very complicated. Just try to invent blot that can increase and
>then magic word "asymptotics".

Ah, I think I got it, but I'm not sure.  To verify my idea, I'd need a
formula for the radius, after one iteration, of a solid circle with an
original radius of R.  This involves trigonometry, and I'm lousy at it.

-- 
Ilmari Karonen - http://www.sci.fi/~iltzu/
"Control: cmsg newgroup sci.math.tasteless"  -- Red Drag Diva in the monastery

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