In this paper, we consider the global comparison problem of Gini means with fixed number of variables on a subinterval I of \(\mathbb {R}_+\) , i.e., the following inequality where \(n\in \mathbb {N},n\ge 2\) is fixed, \((p,q),(r,s)\in \mathbb {R}^2\) and \(x_1,\dots ,x_n\in I\) . Given a nonempty subinterval I of \(\mathbb {R}_+\) and \(n\in \mathbb {N}\) , we introduce the relations \(\begin{aligned} \Gamma _n(I)&:=\{((r,s),(p,q))\in \mathbb {R}^2\times \mathbb {R}^2\mid (*) \text{ holds } \text{ for } \text{ all } x_1,\dots ,x_n\in I\}, \\ \Gamma _\infty (I)&:=\bigcap _{n=1}^\infty \Gamma _n(I). \end{aligned}\) In the paper, we investigate the properties of these sets and their dependence on n and on the interval I and we establish a characterizations of these sets via a constrained minimum problem by using a variant of the Lagrange Multiplier Rule. We also formulate two open problems at the end of the paper.
