A note on equipartition

The problem of the existence of an equi-partition of a curve in R has recently been raised in the context of computational geometry (see [2] and [3]). The problem is to show that for a (continuous) curve Γ : [0, 1] → R and for any positive integer N , there exist points t0 = 0 < t1 < . . . < tN−1 < 1 = tN , such that d(Γ(ti−1),Γ(ti)) = d(Γ(ti),Γ(ti+1)) for all i = 1, . . . , N , where d is a metric or even a semi-metric (a weaker notion) on R. In fact, this problem, for R replaced by any metric space (X, d) was given a positive solution by Urbanik [4] under the (necessary) constraint Γ(0) 6= Γ(1). We show here that the existence of such points, in a broader context, is a consequence of Brower’s fixed point theorem (for reference see any intermediate level book on topology, e.g. [1]).