  The difference set of an outcome in an auction is the set of types that the
auction mechanism maps to the outcome. We give a complete characterization of
the geometry of the difference sets that can appear for a dominant strategy
incentive compatible multi-unit auction showing that they correspond to regular
subdivisions of the unit cube. This observation is then used to construct
mechanisms that are robust in the sense that the set of items allocated to a
player does change only slightly when the player's reported type is changed
slightly.
