  We determine the homeomorphism type of the hyperspace of positively curved
$C^\infty$ convex bodies in $\mathbb R^n$, and derive various properties of its
quotient by the group of Euclidean isometries. We make a systematic study of
hyperspaces of convex bodies that are at least $C^1$. We show how to destroy
the symmetry of a family of convex bodies, and prove that this cannot be done
modulo congruence.
