A ?xed, interval order is considered on a ?nite set of elements. When appropriately de?ned, its representations form a convex polyhedron. Our results describe the geometricstructure of the polyhedron. The facets are in a one-to-one correspondence with the objects of oneof four types: the minimal elements, the contractible elements as well as the noses and the hollowsof the interval order (the latter notions are inferred from Doignon and Falmagne [1997]). Thepolyhedron has only one vertex, which is the minimal representation (in the meaning of Doignon[1988a]; new properties are established here). All representations thus form a convex cone. Wecharacterize the extreme rays of this cone. The uniqueness of the vertex came as a surprise tous surprise because Balof, Doignon and Fiorini [2012] obtained, for the polyhedron formed by allrepresentations of a semiorder, numerous examples with multiple vertices.
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