Hyperspaces of smooth convex bodies up to congruence

We determine the homeomorphism type of the hyperspace of positively curved C∞ convex bodies in Rn, 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 C1. We show how to destroy the symmetry of a family of convex bodies, and prove that this cannot be done modulo congruence.