Given a random variable $X$ and considered a family of its possible distortions, we define two new measures of distance between $X$ and each its distortion. For these distance measures, which are extensions of the Gini's mean difference, conditions are determined for the existence of a minimum, or a maximum, within specific families of distortions, generalizing some results presented in the recent literature.

翻译：给定随机变量 $X$ 并考虑其可能的扭曲族，我们定义了 $X$ 与每个扭曲之间距离的两种新度量。对于这些距离度量（它们是基尼均值差的推广），我们确定了在特定扭曲族内存在最小值或最大值的条件，从而推广了近期文献中的若干结果。