Existence conditions and spreading properties of extreme entropy D-dimensional distributions

The extremization of the information-theoretic measures (Fisher information, Shannon entropy, Tsallis entropy), which complementary describe the spreading of the physical states of natural systems, gives rise to fundamental equations of motion and/or conservation laws. At times, the associated extreme entropy distributions are known for some given constraints, usually moments or radial expectation values. In this work, first we give the existence conditions of the maxent probability distributions in a D-dimensional scenario where two moments (not necessarily of consecutive order) are known. Then we find general relations which involve four elements (the extremized entropy, the other two information-theoretic measures and the variance of the extremum density) in scenarios with different dimensionalities and moment constraints.