# [Banach] Abstract of a paper by M. A. Lopez and S. Reisner

Dale Alspach alspach at www.math.okstate.edu
Tue Jul 31 11:05:53 CDT 2007

This is an announcement for the paper "A note on curves equipartition"
by M. A. Lopez and S. Reisner.

Abstract: The problem of the existence of an equi-partition of a
curve in $\R^n$ has recently been raised in the context of computational
geometry. The problem is to show that for a (continuous) curve
$\Gamma : [0,1] \to \R^n$ and for any positive integer $N$, there
exist points $t_0=0<t_1<...<t_{N-1}<1=t_N$, such that
$d(\Gamma(t_{i-1}),\Gamma(t_i))=d(\Gamma(t_{i}),\Gamma(t_{i+1}))$
for all $i=1,...,N$, where $d$ is a metric or even a semi-metric
(a weaker notion) on $\R^n$. We show here that the existence of
such points, in a much broader context, is a consequence of Brower's
fixed point theorem.

Archive classification: math.FA math.CA

Mathematics Subject Classification: 58C30; 47H10

The source file(s), equipartition.tex: 10551 bytes, is(are) stored
in gzipped form as 0707.4296.gz with size 4kb. The corresponding
postcript file has gzipped size 46kb.

Submitted from: reisner at math.haifa.ac.il

The paper may be downloaded from the archive by web browser from
URL

http://front.math.ucdavis.edu/0707.4296

or

http://arXiv.org/abs/0707.4296

or by email in unzipped form by transmitting an empty message with
subject line

uget 0707.4296

or in gzipped form by using subject line

get 0707.4296

to: math at arXiv.org.