<p>We establish that the asymptotic form of distances within a group of observations as the number of variables tends to infinity is a constant distance matrix. We examine the Euclidean and two dissimilarity matrices, determine their asymptotic representations, and explore their decomposition. An alternative formulation of these indices is proposed, which optimizes computational efficiency. We elucidate the relationship between the eigenvalues of these matrices and established statistical methods for testing the equality of high-dimensional distributions. We demonstrate that the equality of <i>K</i> distributions holds if and only if the corresponding distance matrix exhibits a constant structure. The eigenvalues of the asymptotic dissimilarity matrices are derived for two and three observation groups. Using these limiting forms and eigenvalues, we introduce novel test statistics for assessing the equality of <i>K</i> distributions, drawing inspiration from interpoint distances while circumventing the need for their direct computation.</p>