Hyers–Ulam stability of bijective ε-isometries between Hausdorff metric spaces of compact convex subsets
Let X (resp. Y ) be a real Banach space such that the set of all w ∗ -exposed points of the closed unit ball B ( X ∗ ) (resp. B ( Y ∗ ) ) is w ∗ -dense in the unit sphere S ( X ∗ ) (resp. S ( Y ∗ ) ), ( cc ( X ), H ) (resp. ( cc ( Y ), H )) be the metric space of all nonempty compact convex subset...
Saved in:
| Published in | Aequationes mathematicae Vol. 95; no. 1; pp. 1 - 12 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Cham
Springer International Publishing
01.02.2021
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
Cover
Loading…
| Summary: | Let
X
(resp.
Y
) be a real Banach space such that the set of all
w
∗
-exposed points of the closed unit ball
B
(
X
∗
)
(resp.
B
(
Y
∗
)
) is
w
∗
-dense in the unit sphere
S
(
X
∗
)
(resp.
S
(
Y
∗
)
), (
cc
(
X
),
H
) (resp. (
cc
(
Y
),
H
)) be the metric space of all nonempty compact convex subsets of
X
(resp.
Y
) endowed with the Hausdorff distance
H
, and
f
:
(
c
c
(
X
)
,
H
)
→
(
c
c
(
Y
)
,
H
)
be a standard bijective
ε
-isometry. Then there is a standard surjective isometry
g
:
c
c
(
X
)
→
c
c
(
Y
)
satisfying that
(
1
)
g
|
X
(the restriction of
g
on
{
{
u
}
,
u
∈
X
}
) is a surjective linear isometry from
{
{
u
}
,
u
∈
X
}
onto
{
{
v
}
,
v
∈
Y
}
and
g
(
A
)
=
∪
a
∈
A
g
|
X
(
{
a
}
)
for any
A
∈
c
c
(
X
)
;
(
2
)
H
(
f
(
A
)
,
g
(
A
)
)
≤
3
ε
for any
A
∈
c
c
(
X
)
. |
|---|---|
| ISSN: | 0001-9054 1420-8903 |
| DOI: | 10.1007/s00010-020-00761-y |