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In this paper, we propose a geometric approach to the theory of evidence based on convex geometric interpretations of its two key notions of belief function (b.f.) and Dempster's sum. On one side, we analyze the geometry of b.f.'s as points of a polytope in the Cartesian space called belief space, and discuss the intimate relationship between basic probability assignment and convex combination. On the other side, we study the global geometry of Dempster's rule by describing its action on those convex combinations. By proving that Dempster's sum and convex closure commute, we are able to depict the geometric structure of conditional subspaces, i.e., sets of b.f.'s conditioned by a given function b. Natural applications of these geometric methods to classical problems such as probabilistic approximation and canonical decomposition are outlined.