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

Hyperspaces of smooth convex bodies up to congruence