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On the set of local extrema of a subanalytic function
Let F be a category of subanalytic subsets of real analytic manifolds that is closed under basic set-theoretical operations (locally finite unions, difference and product) and basic topological operations (taking connected components and closures). Let M be a real analytic manifold and denote F ( M...
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| Published in: | Collectanea mathematica (Barcelona) 2020, Vol.71 (1), p.1-24 |
|---|---|
| Main Author: | |
| Format: | Article |
| Language: | English |
| Subjects: | |
| Citations: | Items that this one cites Items that cite this one |
| Online Access: | Get full text |
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| Summary: | Let
F
be a category of subanalytic subsets of real analytic manifolds that is closed under basic set-theoretical operations (locally finite unions, difference and product) and basic topological operations (taking connected components and closures). Let
M
be a real analytic manifold and denote
F
(
M
)
the family of the subsets of
M
that belong to the category
F
. Let
f
:
X
→
R
be a subanalytic function on a subset
X
∈
F
(
M
)
such that the inverse image under
f
of each interval of
R
belongs to
F
(
M
)
. Let
Max
(
f
)
be the set of local maxima of
f
and consider its level sets
Max
λ
(
f
)
:
=
Max
(
f
)
∩
{
f
=
λ
}
=
{
f
=
λ
}
\
Cl
(
{
f
>
λ
}
)
for each
λ
∈
R
. In this work we show that if
f
is continuous, then
Max
(
f
)
=
⨆
λ
∈
R
Max
λ
(
f
)
∈
F
(
M
)
if and only if the family
{
Max
λ
(
f
)
}
λ
∈
R
is locally finite in
M
. If we erase continuity condition, there exist subanalytic functions
f
:
X
→
M
such that
Max
(
f
)
∈
F
(
M
)
, but the family
{
Max
λ
(
f
)
}
λ
∈
R
is not locally finite in
M
or such that
Max
(
f
)
is connected but it is not even subanalytic. We show in addition that if
F
is the category of subanalytic sets and
f
:
X
→
R
is a (non-necessarily continuous) subanalytic map
f
that maps relatively compact subsets of
M
contained in
X
to bounded subsets of
R
, then
Max
(
f
)
∈
F
(
M
)
and the family
{
Max
λ
(
f
)
}
λ
∈
R
is locally finite in
M
. An example of this type of functions are continuous subanalytic functions on closed subanalytic subsets of
M
. The previous results imply that if
F
is either the category of semianalytic sets or the category of
C
-semianalytic sets and
f
is the restriction to an
F
-subset of
M
of an analytic function on
M
, then the family
{
Max
λ
(
f
)
}
λ
∈
R
is locally finite in
M
and
Max
(
f
)
=
⨆
λ
∈
R
Max
λ
(
f
)
∈
F
(
M
)
. We also show that if the category
F
contains the intersections of algebraic sets with real analytic submanifolds and
X
∈
F
(
M
)
is not closed in
M
, then there exists a continuous subanalytic function
f
:
X
→
R
with graph belonging to
F
(
M
×
R
)
such that inverse images under
f
of the intervals of
R
belong to
F
(
M
)
but
Max
(
f
)
does not belong to
F
(
M
)
. As subanalytic sets are locally connected, the set of non-openness points of a continuous subanalytic function
f
:
X
→
R
coincides with the set of local extrema
Extr
(
f
)
:
=
Max
(
f
)
∪
Min
(
f
)
. This means that if
f
:
X
→
R
is a continuous subanalytic function defined on a closed set
X
∈
F
(
M
)
such that the inverse image under
f
of |
|---|---|
| ISSN: | 0010-0757 2038-4815 |
| DOI: | 10.1007/s13348-019-00245-6 |