Vanishing criteria in multiterminal source coding

In Shannon theory literature, nuances of closely related, yet formally dissimilar, vanishing criteria have not been widely studied. For example, the distinctions between the vanishing-distortion and the vanishing-error criteria, and between the vanishing-rate and the zero-rate coding are still not well understood. In this paper, with the help of our recently proposed canonical theory, we show that each pair gives rise to two related geometries, obtained by interchanging the order of section and closure operations on an underlying set. For amenable problems, that set is closed, and hence the rival geometries are indistinguishable. However, we believe that there exists an (intransigent) problem with an underlying open set for which those geometries are distinct. Our belief is buttressed by the fact that the vanishing-error and the zero-error coding problems, involving similarly interchanged section and closure operations, are known to sometimes differ.