Measuring distances in a multidimensional setting is a challenging problem, which appears in many fields of science and engineering. In this paper, to measure the distance between two multivariate distributions, we introduce a new measure of discrepancy which is scale invariant and which, in the case of two independent copies of the same distribution, and after normalization, coincides with the scaling invariant multidimensional version of the Gini index recently proposed in [34]. A byproduct of the analysis is an easy-to-handle discrepancy metric, obtained by application of the theory to a pair of Gaussian multidimensional densities. The obtained metric does improve the standard metrics, based on the mean squared error, as it is scale invariant. The importance of this theoretical finding is illustrated by means of a real problem that concerns measuring the importance of Environmental, Social and Governance factors for the growth of small and medium enterprises.

翻译：在多维环境中度量距离是一个具有挑战性的问题，广泛出现在科学与工程的诸多领域。本文为度量两个多元分布之间的距离，引入了一种新的差异度量方法，该方法具有尺度不变性，并且在处理同一分布的两个独立副本时，经过归一化后，与文献[34]最近提出的尺度不变多维Gini指数版本相一致。该分析的一个副产品是通过将该理论应用于一对高斯多元密度，获得了一种易于处理的差异度量。所获得的度量改进了基于均方误差的标准度量方法，因为它具有尺度不变性。这一理论发现的重要性通过一个实际问题得以阐明，该问题涉及度量环境、社会和治理因素对中小型企业增长的重要性。