Let
G
G
be a compact Lie group,
X
X
a metric
G
G
-space, and
exp
X
\exp X
the hyperspace of all nonempty compact subsets of
X
X
endowed with the Hausdorff metric topology and with the induced action of
G
G
. We prove that the following three assertions are equivalent: (a)
X
X
is locally continuum-connected (resp., connected and locally continuum-connected); (b)
exp
X
\exp X
is a
G
G
-ANR (resp., a
G
G
-AR); (c)
(
exp
X
)
/
G
(\exp X)/G
is an ANR (resp., an AR). This is applied to show that
(
exp
G
)
/
G
(\exp G)/G
is an ANR (resp., an AR) for each compact (resp., connected) Lie group
G
G
. If
G
G
is a finite group, then
(
exp
X
)
/
G
(\exp X)/G
is a Hilbert cube whenever
X
X
is a nondegenerate Peano continuum. Let
L
(
n
)
L(n)
be the hyperspace of all centrally symmetric, compact, convex bodies
A
⊂
R
n
A\subset \mathbb {R}^n
,
n
≥
2
n\ge 2
, for which the ordinary Euclidean unit ball is the ellipsoid of minimal volume containing
A
A
, and let
L
0
(
n
)
L_0(n)
be the complement of the unique
O
(
n
)
O(n)
-fixed point in
L
(
n
)
L(n)
. We prove that: (1) for each closed subgroup
H
⊂
O
(
n
)
H\subset O(n)
,
L
0
(
n
)
/
H
L_0(n)/H
is a Hilbert cube manifold; (2) for each closed subgroup
K
⊂
O
(
n
)
K\subset O(n)
acting non-transitively on
S
n
−
1
S^{n-1}
, the
K
K
-orbit space
L
(
n
)
/
K
L(n)/K
and the
K
K
-fixed point set
L
(
n
)
[
K
]
L(n)[K]
are Hilbert cubes. As an application we establish new topological models for tha Banach-Mazur compacta
L
(
n
)
/
O
(
n
)
L(n)/O(n)
and prove that
L
0
(
n
)
L_0(n)
and
(
exp
S
n
−
1
)
∖
{
S
n
−
1
}
(\exp S^{n-1})\setminus \{S^{n-1}\}
have the same
O
(
n
)
O(n)
-homotopy type.