A note on equipartition
Abstract
Description
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 $Γ: [0,1] \to \R^n$ and for any positive integer N, there exist points $t_0=0<t_1<...<t_{N-1}<1=t_N$, such that $d(Γ(t_{i-1}),Γ(t_i))=d(Γ(t_{i}),Γ(t_{i+1}))$ 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 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
Some misprints in earlier versions are corrected, one reference is added with remarks concerning it