On Shortest Polygonal Approximations to a Curve

Abstract

Let F be a set consisting of n points of a metric space. If ${{F}^{P}} = [{{p}_{1}},{{p}_{2}}, \ldots ,{{p}_{n}}]$ is a polygon (i.e., a finite ordered set) consisting of the points of F, we set $1({{F}^{P}}) = \sum {{p}_{i}}{{p}_{{i + 1}}}$, where p_ip_i+1 denotes the distance from p_i to p_i+1. The smallest of the n! numbers 1(F^Q) formed for the n! permutations Q of the numbers 1, 2,…, n will be denoted by λ(F). Thus λ(F) is the length of the shortest polygon that can be inscribed into F.