Geodesic
Topology of the hyperspace of convex bodies of constant width
Matematičeskie zametki, Tome 62 (1997) no. 6, pp. 813-819

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The hyperspace of all convex bodies of constant width in Euclidean space $\mathbb R^n$, $n\ge2$, is proved to be homeomorphic to a contractible $Q$-manifold ($Q$ denotes the Hilbert cube). The proof makes use of an explicitly constructed retraction of the entire hyperspace of convex bodies on the hyperspace of convex bodies of constant width. 
@article{MZM_1997_62_6_a1,
     author = {L. E. Bazilevich},
     title = {Topology of the hyperspace of convex bodies of constant width},
     journal = {Matemati\v{c}eskie zametki},
     pages = {813--819},
     publisher = {mathdoc},
     volume = {62},
     number = {6},
     year = {1997},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1997_62_6_a1/}
}
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L. E. Bazilevich. Topology of the hyperspace of convex bodies of constant width. Matematičeskie zametki, Tome 62 (1997) no. 6, pp. 813-819. http://geodesic.mathdoc.fr/item/MZM_1997_62_6_a1/