From: rusin@math.niu.edu (Dave Rusin)
Newsgroups: sci.math.research
Subject: Re: Volume and Surface areas of Hyperspheres
Date: 16 Jan 1998 17:32:33 GMT
In article <34BB6663.40861919@intermetrics.com>,
Ron Kohl wrote:
>Speaking of hyperspheres and volumes, I am wondering if anyone can
>provide a reference to theorem that I recall from long ago during my
>grad school days called, "The Ham Sandwich Theorem". It stated
>something like:
>Given two pieces of bread and a slice of ham, I can 'assemble' a
>sandwich of these 3 items (by placing the 3 items anywhere in 3-space!!)
>and there always exists a single swip of a knife (who's trace is a
>hyperplane) such that all 3 pieces will be equally divided.
[many pointless bytes of original post deleted.]
Right. State it carefully so that it's clear that
"...for every assembly of the sandwich there exists..."
rather than this possible interpretation of your statement
"...for some assembly of the sandwich there exists..."
The pieces of your sandwich need only be bounded and measurable.
I learned this as the Borsuk-Ulam theorem, but in Fulton's algebraic
topology text he lists this as a corollary to the B-U theorem which
he attributes to Stone and Tukey (presumably the article in
Duke Math Jour 9 (1942) 356--359 ).
dave
[sensing that sci.math.research is now definitely sci.math.graduate.moderated]