From: lrudolph@panix.com (Lee Rudolph)
Subject: Re: Borsuk-Ulam Theorem/Ham Sandwiches
Date: 18 Jan 2000 19:08:32 -0500
Newsgroups: sci.math
Summary: [missing]

Fred Galvin <galvin@math.ukans.edu> writes:

>On 18 Jan 2000, Francesca Sanderson wrote:
>
>> Does anyone know of any implications of the Borsuk-Ulam theorem other
>> than the Ham Sandwich theorem and the Brouwer fixed-point theorem? I
>> am trying to flesh out a project which is currently looking a little
>> basic, and is in need of interesting and innovative results. Any
>> comments on the Ham Sandwich theorem/Stone and Tukey's 1942 Duke paper
>> 'Generalized Sandwich Theorems' would also be welcomed. 
>
>For an application to graph theory, see I. Barany, A short proof of
>Kneser's conjecture, J. Combin. Theory Ser. A 25 (1978), 325-326.

MathSciNet turned up 26 matches for "ham sandwich".  One of them
is the following.

--begin copyright violation--
95g:52007 
Barany, Imre(H-AOS) 
Geometric and combinatorial applications of Borsuk's theorem. 
New trends in discrete and computational geometry, 235--249, 
Algorithms Combin., 10, Springer, Berlin, 1993. 

Probably the first really striking fact discovered in topology after 
the initial contributions of Poincare (1892--1904) was the Borsuk-Ulam 
theorem (1933): There is no continuous map from a sphere to a lower-
dimensional sphere which commutes with their antipodal involutions. 
This result, and its many variants and generalizations, have numerous 
applications. In this very nice survey the author lists some of the 
geometrical and combinatorial applications which have been recently 
made of this theorem. He also gives, in sketch form, a clear idea of 
some arguments. The topics covered include theorems of the 
van Kampen-Flores-Tverberg kind, generalized "pigeon-hole" theorems 
of Kneser-Lovasz-Erdos type, generalizations of the "ham-sandwich 
theorem" due to Zivaljevic-Vrecica and others, and some results 
regarding sphere coverings and face numbers of centrally symmetric 
polytopes. The author has himself obtained very elegant results in 
almost all of these fields. Using one such result, the reviewer has 
given [Israel J. Math. 79 (1992), no. 2-3, 317--320; MR 94j:52011] 
a new, short and conceptual proof of Tverberg's theorem, which, 
together with the method of deleted joins, should lead to interesting 
new developments in this field. 

Reviewed by K. S. Sarkaria 

--end copyright violation--

Lee Rudolph