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The reducing transformation and global optimization technique called Alienor has been developed in the 1980s by Cherruault and Guillez. These methods are based on the approximating properties of α ‐dense curves. The aim of this work is to give a very large class of functions generating α ‐dense curves in a hyper‐rectangle of Rn.

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Some results concerning the existence of -dense curves with minimal length are given. This type of curves used in the reducing transformation called Alienor was invented by Cherruault and Guillez. They have been applied to global optimization in the following way: a multivariable optimization problem is transformed in an optimization problem depending on a single variable. Then this idea was extended by Cherruault and his team for obtaining general classes of reducing transformations having minimal properties (length of the -dense curves, minimization of the calculus time, etc.).