A note on equipartition

Abstract

The problem of the existence of an equi-partition of a curve in ${\mathbb{R}}^n$ has recently been raised in the context of computational geometry. The problem is to show that for a (continuous) curve $\Gamma :[0,1]\to {\mathbb{R}}^n$ 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 ${\mathbb{R}}^n$. We show here that the existence of such points, in a broader context, is a consequence of Brower's fixed point theorem.

Some misprints in earlier versions are corrected, one reference is added with remarks concerning it