  The semigroup of convex bodies in ${\mathbb R}^n$ with Minkowski addition has
a canonical embedding into an abelian group; its elements have been called
virtual convex bodies. Geometric interpretations of such virtual convex bodies
have been particularly fruitful under the restriction to polytopes. For general
convex bodies, mainly the planar case has been studied, as a part of the more
general investigation of hedgehogs. Here we restrict ourselves to strictly
convex bodies in ${\mathbb R}^n$. A particularly natural geometric
interpretation of virtual convex bodies can then be seen in the set of
differences of boundary points of two convex bodies with the same outer normal
vector. We describe how in the typical case (in the sense of Baire category)
this leads to a highly singular object.
