<p>Diversity measures are essential tools for ecological and environmental assessments, particularly in the context of increasingly complex and large-scale datasets. We discuss statistical properties of diversity measures such as the Gini-Simpson index, the Hill number, Rao’s quadratic entropy and the Leinster-Cobbold index. We explore the distribution maximizing such a diversity measure under linear constraints that reflect ecological realities, such as resource competition or habitat suitability. Furthermore, we discuss the information geometry associated with the maximum diversity distribution focusing on the cross diversity measures such as the cross entropy. This gives an explicit geodesic characterization of Hill number maximizing distribution under linear constraints.</p>