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Re: Balance point of a solid

• To: mathgroup at smc.vnet.net
• Subject: [mg113781] Re: Balance point of a solid
• From: Ray Koopman <koopman at sfu.ca>
• Date: Fri, 12 Nov 2010 05:27:50 -0500 (EST)
• References: <iam332$jvn$1@smc.vnet.net> <ibgj10$ec3$1@smc.vnet.net>

On Nov 11, 3:11 am, Andreas <aa... at ix.netcom.com> wrote:
> Daniel, Ray, Clifford -- Many thanks for the thought provoking
> contributions.
>
> Ray and others have found using integration on this problem takes
> inordinately long to calculate once you get to 5 dimensions.
>
> Could one attack this problem in another way?  It occurred to me
> that given that we know the lengths of the base simplex and heights
> of the trapezoids as well as the right angles of the heights to the
> base simplex one could then calculate the length of the top lines and
> solve the entire thing geometrically without needing to integrate.
> Not necessarily pretty or elegant but it might give give a solution
> that calculates fast.
>
> Anyone think this could work?

The solution I found, rewritten here in the notation of my Nov 5
post as  ((h/Tr@h + 1)/(1 + Length@h)).A ,  is fast and agrees
with the results given by integrating. What's missing is a proof.