A note on equipartition

TL;DR

This paper investigates the existence of equi-partitions of curves in Euclidean space, showing that such partitions can be established using Brouwer's fixed point theorem, thus connecting geometric partition problems with topological fixed point results.

Contribution

It demonstrates that the existence of equi-partitions for curves in  space follows from Brouwer's fixed point theorem, broadening the understanding of geometric partition problems.

Findings

Existence of equi-partitions proven using Brouwer's fixed point theorem.

Applicable to curves with metrics or semi-metrics.

Provides a topological approach to a geometric problem.

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 broader context, is a consequence of Brower's fixed point theorem.