On the minimal length curve that densifies the square Available to Purchase

This paper deals with the existence of a curve of minimal length, expressed in parametric coordinates, which densifies the square J²=≤ft [ −1,1\right ] × ≤ft [ −1,1\right ] \ with a given degree of density α. Nevertheless, the same problem has no solution if we consider the family of curves defined by means of the graphics of continuous and rectifiable functions f: J→ J. Their consequences on the approximation method to the global optimization are also derived.