LetGGbe a compact Lie group,XXa metricGG-space, andexpX\exp Xthe hyperspace of all nonempty compact subsets ofXXendowed with the Hausdorff metric topology and with the induced action ofGG. We prove that the following three assertions are equivalent: (a)XXis locally continuum-connected (resp., connected and locally continuum-connected); (b)expX\exp Xis aGG-ANR (resp., aGG-AR); (c)(expX)/G(\exp X)/Gis an ANR (resp., an AR). This is applied to show that(expG)/G(\exp G)/Gis an ANR (resp., an AR) for each compact (resp., connected) Lie groupGG. IfGGis a finite group, then(expX)/G(\exp X)/Gis a Hilbert cube wheneverXXis a nondegenerate Peano continuum. LetL(n)L(n)be the hyperspace of all centrally symmetric, compact, convex bodiesA⊂RnA\subset \mathbb {R}^n,n≥2n\ge 2, for which the ordinary Euclidean unit ball is the ellipsoid of minimal volume containingAA, and letL0(n)L_0(n)be the complement of the uniqueO(n)O(n)-fixed point inL(n)L(n). We prove that: (1) for each closed subgroupH⊂O(n)H\subset O(n),L0(n)/HL_0(n)/His a Hilbert cube manifold; (2) for each closed subgroupK⊂O(n)K\subset O(n)acting non-transitively onSn−1S^{n-1}, theKK-orbit spaceL(n)/KL(n)/Kand theKK-fixed point setL(n)[K]L(n)[K]are Hilbert cubes. As an application we establish new topological models for tha Banach-Mazur compactaL(n)/O(n)L(n)/O(n)and prove thatL0(n)L_0(n)and(expSn−1)∖{Sn−1}(\exp S^{n-1})\setminus \{S^{n-1}\}have the sameO(n)O(n)-homotopy type.