Hyperplane cutting up objects

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Talith
Proved the Goldbach Conjecture
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Hyperplane cutting up objects

Postby Talith » Fri Jun 12, 2009 1:57 am UTC

It's rare that my little brother inspires mathematical insights in me but today was I had precisely one of those moments. We were cutting rolls in half for a party and were getting pretty bored so my brother said, "isn't there a way to cut all of these rolls in half with just one cut?" and ofcourse I said it's impossible to do unless we move the rolls but I pointed out that if you have 2 rolls arbitrarily placed on the plane of the table, there always exists a cut which will bisect both rolls by volume which would be fairly simple to prove using the intermediate value theorem. It got me thinking though, what happens if we bump up the dimensions? In R^3 it seems logical that you would be able to cut 3 objects (by volume) with a plane (this in itself seems pretty difficult to prove) and as the dimensions increase one would expect that n objects in R^n can be cut with a single (n-1) dimensional hyperplane. I wonder if anyone can offer any insight on proving the last statement (or even just the R^3 statement) because although i can kind of picture planes sweeping through higher dimensional spaces, putting it into words and formalising it is out of my reach at the moment :P.

hnooch
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Re: Hyperplane cutting up objects

Postby hnooch » Fri Jun 12, 2009 2:02 am UTC

http://en.wikipedia.org/wiki/Ham_sandwich_theorem

Yup, your intuition is right, but it's not exactly trivial.

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Talith
Proved the Goldbach Conjecture
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Joined: Sat Nov 29, 2008 1:28 am UTC
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Re: Hyperplane cutting up objects

Postby Talith » Fri Jun 12, 2009 2:30 am UTC

Haha thanks, I don't think you could have found a page more apt (god bless wikipedia) also I love the name of the theorem, really appreciate it, the proof looks okish, I'll work my head around it when I've had a bit more sleep.


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