Analysis of a shape descriptor: Distance between two shape centroids versus shape diameter

In this paper we study a shape descriptor ρ(S) that is defined as the ratio of the squared distance between centroids and the squared diameter of shape S (i.e., a set in the plane). This descriptor has been discussed for more than ten years, but its behaviour is not yet well studied, thus hindering its wide-spread application. There are two open basic questions to be answered: 1. What is the range of ρ(S)? 2. How do shapes look like with large ρ(S) values? This paper answers both open questions. We show that ρ(S) values are in the interval [0; 1), meaning in particular that value 1 is not taken, that 1 is the best possible upper bound, and and we give examples of shapes whose ρ(S) values are arbitrarily close to 1.