The normal distribution in some constrained sample spaces

Phenomena with a constrained sample space appear frequently in practice. This is the case e.g. with strictly positive data and with compositional data, like percentages and the like. If the natural measure of difference is not the absolute one, it is possible to use simple algebraic properties to show that it is more convenient to work with a geometry that is not the usual Euclidean geometry in real space, and with a measure which is not the usual Lebesgue measure, leading to alternative models which better fit the phenomenon under study. The general approach is presented and illustrated both on the positive real line and on the D-part simplex.