Necessary conditions for dominant strategy implementability on a restricted type space are identified for a finite set of alternatives.  For any one-person mechanism obtained by fixing the other individuals' types, the geometry of the partition of the type space into subsets that are allocated the same alternative is analyzed using difference set polyhedra.  Situations are identified in which it is necessary for all cycle lengths in the corresponding allocation graph to be zero, which is shown to be equivalent to the vertices of the difference sets restricted to normalized type vectors coinciding.  For an arbitrary type space, it is also shown that any one-person dominant strategy implementable allocation function (i) can be extended to the unrestricted domain and (ii) that it is the solution to an affine maximization problem.