Let
X
X
be a Peano continuum. Let
2
x
{2^x}
(resp.,
C
(
X
)
C(X)
) be the space of all nonempty compacta (resp., subcontinua) of
X
X
with the Hausdorff matric. Let
ω
\omega
be a Whitney map defined on
H
=
2
X
\mathcal {H}={2^{X}}
or
C
(
X
)
C(X)
such that
ω
\omega
is admissible (this requires the existence of a certain type of deformation of
H
\mathcal {H}
). If
H
=
C
(
X
)
\mathcal {H}=C(X)
, assume
X
X
contains no free arc. Then, for any
t
0
∈
(
0
,
ω
(
X
)
)
{t_0} \in (0,\omega (X))
, it is proved that
ω
−
1
(
t
0
)
,
ω
−
1
(
[
0
,
t
0
]
)
{\omega ^{ - 1}}({t_0}),\,{\omega ^{ - 1}}([0,\,{t_0}])
, and
ω
−
1
(
[
t
0
,
ω
(
X
)
]
)
{\omega ^{ - 1}}([{t_0},\,\omega (X)])
are Hilbert cubes. This is an analogue of the Curtis-Schori theorem for
H
\mathcal {H}
. A general result for the existance of admissible Whitney maps is proved which implies that these maps exist when
X
X
is starshaped in a Banach space or when
X
X
is a dendrite. Using these results it is shown, for example that being an AR, an ANR, an LC space, or an
L
C
n
{\text {L}}{{\text {C}}^n}
space is not strongly Whitney-reversible.