Measuring inequality in high dimensions: a Gini-based approach

While classical inequality indices–such as the Gini and pq-mean indices–satisfy a set of six desirable axiomatic properties in the univariate case (Hurley and Rickard. Comparing measures of sparsity. IEEE Trans Inf Theory. 2009;55(10):4723–4741. doi: 10.1109/TIT.2009.2027527), extending these properties to the multivariate context poses a significant challenge, primarily due to the absence of a natural ordering in higher dimensions. Previous multivariate generalizations, although mathematically elegant, often lack interpretability and fail to preserve scale invariance. We analyze the approach proposed in Auricchio et al. (Extending the Gini index to higher dimensions via whitening processes. Rend Lincei. 2025;35(3):511–528), demonstrating that, under suitable conditions, it yields a multivariate Gini index that satisfies all six axioms and offers a practical, interpretable framework for measuring the unevenness of high-dimensional data.