Geometry and Combinatorics of Plausibility and Commonality Functions

In this work we extend the geometric approach to the theory of evidence in order to study the geometric behavior of the two quantities inherently associated with a belief function. i.e. the plausibility and commonality functions. After introducing the analogous of the basic probability assignment for plausibilities and commonalities, we exploit it to understand the simplicial form of both plausibility and commonality spaces. Given the intuition provided by the binary case we prove the congruence of belief, plausibility, and commonality spaces for both standard and unnormalized belief functions, and describe the point-wise geometry of these sum functions in terms of the rigid transformation mapping them onto each other. This leads us to conjecture that the D-S formalism may be in fact a geometric calculus in the line of geometric probability, and opens the way to a wider application of discrete mathematics to subjective probability.