Measuring distances in multidimensional settings poses a significant challenge encountered across various scientific and engineering disciplines. In this paper, we introduce a novel measure of divergence to quantify the discrepancy between two multidimensional distributions—one predicted by a machine learning model and the other expected. Our approach builds upon the class of Energy Distances and incorporates a whitening pre-processing step, resulting in a divergence that is strictly connected to the new multivariate Gini index. To validate the proposed divergence, we demonstrate its effectiveness as a loss function for training a neural network designed to predict the financial performance of small and medium enterprises.

We show that Shannon's entropy-power inequality admits a strengthened version in the case in which the densities are log-concave. In such a case, in fact, one can extend the Blachman-Stam argument to obtain a sharp inequality for the second derivative of Shannon's entropy functional with respect to the heat semigroup.

We associate to the pth Rényi entropy a definition of entropy power, which is the natural extension of Shannon's entropy power and exhibits a nice behavior along solutions to the p-nonlinear heat equation in Rn. We show that the Rényi entropy power of general probability densities solving such equations is always a concave function of time, whereas it has a linear behavior in correspondence to the Barenblatt source-type solutions. This result extends Costa's concavity inequality for Shannon's entropy power to Rényi entropies.