# Hyperspaces of smooth convex bodies up to congruence

## TL;DR

This paper characterizes the topological structure of the space of smooth convex bodies with positive curvature in Euclidean space and explores symmetry properties and quotient spaces under Euclidean isometries.

## Contribution

It determines the homeomorphism type of the hyperspace of positively curved smooth convex bodies and studies symmetry-breaking in convex body families.

## Key findings

- The hyperspace of positively curved smooth convex bodies has a specific topological type.
- Symmetry of convex body families cannot be destroyed modulo congruence.
- Various properties of the quotient space under Euclidean isometries are derived.

## Abstract

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.