σ-Porosity of the set of strict contractions in a space of
non-expansive mappings
Christian Bargetz
joint work with Michael Dymond
Relations Between Banach Space Theory
and Geometric Measure Theory
8–12 June 2015
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
1 / 12
The setting
Let X be a Banach space and C ⊂ X a closed, convex and bounded set.
We consider the space
M = {f : C → C : ∀x, y ∈ C : kf (x) − f (y )k ≤ kx − y k}
equipped with the metric
d(f , g ) = sup kf (x) − g (x)k
x∈C
of uniform convergence.
M is a complete metric space.
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
2 / 12
The setting
Let X be a Banach space and C ⊂ X a closed, convex and bounded set.
We consider the space
M = {f : C → C : ∀x, y ∈ C : kf (x) − f (y )k ≤ kx − y k}
equipped with the metric
d(f , g ) = sup kf (x) − g (x)k
x∈C
of uniform convergence.
M is a complete metric space.
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
2 / 12
The setting
Let X be a Banach space and C ⊂ X a closed, convex and bounded set.
We consider the space
M = {f : C → C : ∀x, y ∈ C : kf (x) − f (y )k ≤ kx − y k}
equipped with the metric
d(f , g ) = sup kf (x) − g (x)k
x∈C
of uniform convergence.
M is a complete metric space.
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
2 / 12
How small is the set of strict contrations?
We consider the set
N = {f ∈ M : Lip(f ) < 1}
of strict contractions.
Question
“How small” is the set of strict contractions in M?
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
3 / 12
How small is the set of strict contrations?
We consider the set
N = {f ∈ M : Lip(f ) < 1}
of strict contractions.
Question
“How small” is the set of strict contractions in M?
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
3 / 12
σ-porous sets
A subset A ⊂ M is said to be porous at x ∈ A if there are constants α > 0
and ε0 > 0 with the following property: For all ε ∈ (0, ε0 ) there is a point
y ∈ M with ky − xk ≤ ε and B(y , α ε) ∩ A = ∅. The set A is called
porous if it is porous at all of its points.
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
4 / 12
σ-porous sets
A subset A ⊂ M is said to be porous at x ∈ A if there are constants α > 0
and ε0 > 0 with the following property: For all ε ∈ (0, ε0 ) there is a point
y ∈ M with ky − xk ≤ ε and B(y , α ε) ∩ A = ∅. The set A is called
porous if it is porous at all of its points.
Picture
x
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
4 / 12
σ-porous sets
A subset A ⊂ M is said to be porous at x ∈ A if there are constants α > 0
and ε0 > 0 with the following property: For all ε ∈ (0, ε0 ) there is a point
y ∈ M with ky − xk ≤ ε and B(y , α ε) ∩ A = ∅. The set A is called
porous if it is porous at all of its points.
Picture
x
ε0
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
4 / 12
σ-porous sets
A subset A ⊂ M is said to be porous at x ∈ A if there are constants α > 0
and ε0 > 0 with the following property: For all ε ∈ (0, ε0 ) there is a point
y ∈ M with ky − xk ≤ ε and B(y , α ε) ∩ A = ∅. The set A is called
porous if it is porous at all of its points.
Picture
x
ε0
ε
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
4 / 12
σ-porous sets
A subset A ⊂ M is said to be porous at x ∈ A if there are constants α > 0
and ε0 > 0 with the following property: For all ε ∈ (0, ε0 ) there is a point
y ∈ M with ky − xk ≤ ε and B(y , α ε) ∩ A = ∅. The set A is called
porous if it is porous at all of its points.
Picture
x
αε
ε0
ε
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
4 / 12
σ-porous sets
A subset A ⊂ M is said to be porous at x ∈ A if there are constants α > 0
and ε0 > 0 with the following property: For all ε ∈ (0, ε0 ) there is a point
y ∈ M with ky − xk ≤ ε and B(y , α ε) ∩ A = ∅. The set A is called
porous if it is porous at all of its points.
Picture
x
ε
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
4 / 12
σ-porous sets
A subset A ⊂ M is said to be porous at x ∈ A if there are constants α > 0
and ε0 > 0 with the following property: For all ε ∈ (0, ε0 ) there is a point
y ∈ M with ky − xk ≤ ε and B(y , α ε) ∩ A = ∅. The set A is called
porous if it is porous at all of its points.
Picture
αε
x
ε
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
4 / 12
σ-porous sets
A subset A ⊂ M is said to be porous at x ∈ A if there are constants α > 0
and ε0 > 0 with the following property: For all ε ∈ (0, ε0 ) there is a point
y ∈ M with ky − xk ≤ ε and B(y , α ε) ∩ A = ∅. The set A is called
porous if it is porous at all of its points.
Picture
x
ε
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
4 / 12
σ-porous sets
A subset A ⊂ M is said to be porous at x ∈ A if there are constants α > 0
and ε0 > 0 with the following property: For all ε ∈ (0, ε0 ) there is a point
y ∈ M with ky − xk ≤ ε and B(y , α ε) ∩ A = ∅. The set A is called
porous if it is porous at all of its points.
Picture
αε x
ε
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
4 / 12
σ-porous sets
A subset A ⊂ M is said to be porous at x ∈ A if there are constants α > 0
and ε0 > 0 with the following property: For all ε ∈ (0, ε0 ) there is a point
y ∈ M with ky − xk ≤ ε and B(y , α ε) ∩ A = ∅. The set A is called
porous if it is porous at all of its points.
Picture
x
ε
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
4 / 12
σ-porous sets
A subset A ⊂ M is said to be porous at x ∈ A if there are constants α > 0
and ε0 > 0 with the following property: For all ε ∈ (0, ε0 ) there is a point
y ∈ M with ky − xk ≤ ε and B(y , α ε) ∩ A = ∅. The set A is called
porous if it is porous at all of its points.
Picture
x
ε
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
4 / 12
σ-porous sets
A subset A ⊂ M is said to be porous at x ∈ A if there are constants α > 0
and ε0 > 0 with the following property: For all ε ∈ (0, ε0 ) there is a point
y ∈ M with ky − xk ≤ ε and B(y , α ε) ∩ A = ∅. The set A is called
porous if it is porous at all of its points.
Picture
x
ε
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
4 / 12
σ-porous sets
A subset A ⊂ M is said to be porous at x ∈ A if there are constants α > 0
and ε0 > 0 with the following property: For all ε ∈ (0, ε0 ) there is a point
y ∈ M with ky − xk ≤ ε and B(y , α ε) ∩ A = ∅. The set A is called
porous if it is porous at all of its points.
Picture
x
ε
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
4 / 12
σ-porous sets
A subset A ⊂ M is said to be porous at x ∈ A if there are constants α > 0
and ε0 > 0 with the following property: For all ε ∈ (0, ε0 ) there is a point
y ∈ M with ky − xk ≤ ε and B(y , α ε) ∩ A = ∅. The set A is called
porous if it is porous at all of its points.
Picture
x
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
4 / 12
σ-porous sets
A subset A ⊂ M is said to be porous at x ∈ A if there are constants α > 0
and ε0 > 0 with the following property: For all ε ∈ (0, ε0 ) there is a point
y ∈ M with ky − xk ≤ ε and B(y , α ε) ∩ A = ∅. The set A is called
porous if it is porous at all of its points.
The set A is called σ-porous if it is a countable union of porous sets.
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
4 / 12
σ-porous sets
A subset A ⊂ M is said to be porous at x ∈ A if there are constants α > 0
and ε0 > 0 with the following property: For all ε ∈ (0, ε0 ) there is a point
y ∈ M with ky − xk ≤ ε and B(y , α ε) ∩ A = ∅. The set A is called
porous if it is porous at all of its points.
The set A is called σ-porous if it is a countable union of porous sets.
Note that σ-porous sets are of first category in the sense of the Baire
category theorem.
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
4 / 12
The Hilbert space case
Theorem (De Blasi and Myjak, 1989)
If X is a Hilbert space the set N of strict contractions on C is a σ-porous
subset of M.
Proof sketch.
Take a sequence (Lk )k∈N with Ln % 1 and set
Nk = {f ∈ M : Lip(f ) ≤ Lk }.
Given f ∈ Nk and ε > 0 set g to the identity on a small ball around the
fixed point x ∗ of f and to f outside a bigger ball around x ∗ then use
Kirszbraun’s extension theorem to get a close enough midpoint of a ball
outside Nk .
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
5 / 12
The Hilbert space case
Theorem (De Blasi and Myjak, 1989)
If X is a Hilbert space the set N of strict contractions on C is a σ-porous
subset of M.
Proof sketch.
Take a sequence (Lk )k∈N with Ln % 1 and set
Nk = {f ∈ M : Lip(f ) ≤ Lk }.
Given f ∈ Nk and ε > 0 set g to the identity on a small ball around the
fixed point x ∗ of f and to f outside a bigger ball around x ∗ then use
Kirszbraun’s extension theorem to get a close enough midpoint of a ball
outside Nk .
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
5 / 12
The Hilbert space case
Theorem (De Blasi and Myjak, 1989)
If X is a Hilbert space the set N of strict contractions on C is a σ-porous
subset of M.
Proof sketch.
Take a sequence (Lk )k∈N with Ln % 1 and set
Nk = {f ∈ M : Lip(f ) ≤ Lk }.
Given f ∈ Nk and ε > 0 set g to the identity on a small ball around the
fixed point x ∗ of f and to f outside a bigger ball around x ∗ then use
Kirszbraun’s extension theorem to get a close enough midpoint of a ball
outside Nk .
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
5 / 12
The general case
Theorem (B. and Dymond, 2015)
Let X be a Banach space and C ⊂ X a closed, convex and bounded set.
Then the set N of all strict contractions is a σ-porous subset of M.
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
6 / 12
Sketch of the proof
C
0
S(X )
We define
p
Na,b
1
= f ∈ N : a < Lip(f , Γ) ≤ b, Lip(f ) ≤ 1 −
.
p
p
Fix f ∈ Na,b
. Choose x0 ∈ Γ such that
lim inf
t→0+
Christian Bargetz (University of Innsbruck)
kf (x0 + te) − f (x0 )k
≥ a.
t
σ-Porosity of the set of strict contractions
11 June 2015
7 / 12
Sketch of the proof
C
0
S(X )
We define
p
Na,b
1
= f ∈ N : a < Lip(f , Γ) ≤ b, Lip(f ) ≤ 1 −
.
p
p
Fix f ∈ Na,b
. Choose x0 ∈ Γ such that
lim inf
t→0+
Christian Bargetz (University of Innsbruck)
kf (x0 + te) − f (x0 )k
≥ a.
t
σ-Porosity of the set of strict contractions
11 June 2015
7 / 12
Sketch of the proof
C
0
S(X )
We define
p
Na,b
1
= f ∈ N : a < Lip(f , Γ) ≤ b, Lip(f ) ≤ 1 −
.
p
p
Fix f ∈ Na,b
. Choose x0 ∈ Γ such that
lim inf
t→0+
Christian Bargetz (University of Innsbruck)
kf (x0 + te) − f (x0 )k
≥ a.
t
σ-Porosity of the set of strict contractions
11 June 2015
7 / 12
Sketch of the proof
C
e
Γ
0
S(X )
We define
p
Na,b
1
= f ∈ N : a < Lip(f , Γ) ≤ b, Lip(f ) ≤ 1 −
.
p
p
Fix f ∈ Na,b
. Choose x0 ∈ Γ such that
lim inf
t→0+
Christian Bargetz (University of Innsbruck)
kf (x0 + te) − f (x0 )k
≥ a.
t
σ-Porosity of the set of strict contractions
11 June 2015
7 / 12
Sketch of the proof
C
e
Γ
0
S(X )
We define
p
Na,b
1
= f ∈ N : a < Lip(f , Γ) ≤ b, Lip(f ) ≤ 1 −
.
p
p
Fix f ∈ Na,b
. Choose x0 ∈ Γ such that
lim inf
t→0+
Christian Bargetz (University of Innsbruck)
kf (x0 + te) − f (x0 )k
≥ a.
t
σ-Porosity of the set of strict contractions
11 June 2015
7 / 12
Sketch of the proof
C
e
Γ
0
S(X )
We define
p
Na,b
1
= f ∈ N : a < Lip(f , Γ) ≤ b, Lip(f ) ≤ 1 −
.
p
p
Fix f ∈ Na,b
. Choose x0 ∈ Γ such that
lim inf
t→0+
Christian Bargetz (University of Innsbruck)
kf (x0 + te) − f (x0 )k
≥ a.
t
σ-Porosity of the set of strict contractions
11 June 2015
7 / 12
Sketch of the proof
C
e
Γ
0
S(X )
x0
We define
p
Na,b
1
= f ∈ N : a < Lip(f , Γ) ≤ b, Lip(f ) ≤ 1 −
.
p
p
Fix f ∈ Na,b
. Choose x0 ∈ Γ such that
lim inf
t→0+
Christian Bargetz (University of Innsbruck)
kf (x0 + te) − f (x0 )k
≥ a.
t
σ-Porosity of the set of strict contractions
11 June 2015
7 / 12
Sketch of the proof, part II
C
Γ
0
x0
Fix α > 0, ε > 0. Now set
g (x) = f (x + σφε (e ∗ (x − x0 ))(e − (x − x0 ))).
where σφε (e ∗ (x − x0 ))(e − (x − x0 )) stretches along Γ to increase the
Lipschitz constant.
Setting R = diam(C ), if b − a is small enough, we obtain
g ∈M
d(f , g ) ≤ Rε
p
B(g , αRε) ∩ Na,b
= ∅.
p
Hence Na,b
is porous.
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
8 / 12
Sketch of the proof, part II
C
Γ
0
x0
Fix α > 0, ε > 0. Now set
g (x) = f (x + σφε (e ∗ (x − x0 ))(e − (x − x0 ))).
where σφε (e ∗ (x − x0 ))(e − (x − x0 )) stretches along Γ to increase the
Lipschitz constant.
Setting R = diam(C ), if b − a is small enough, we obtain
g ∈M
d(f , g ) ≤ Rε
p
B(g , αRε) ∩ Na,b
= ∅.
p
is porous.
Hence Na,b
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
8 / 12
Sketch of the proof, part II
C
Γ
0
x0
Fix α > 0, ε > 0. Now set
g (x) = f (x + σφε (e ∗ (x − x0 ))(e − (x − x0 ))).
where σφε (e ∗ (x − x0 ))(e − (x − x0 )) stretches along Γ to increase the
Lipschitz constant.
Setting R = diam(C ), if b − a is small enough, we obtain
g ∈M
d(f , g ) ≤ Rε
p
B(g , αRε) ∩ Na,b
= ∅.
p
is porous.
Hence Na,b
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
8 / 12
Sketch of the proof, part II
C
Γ
0
x0
Fix α > 0, ε > 0. Now set
g (x) = f (x + σφε (e ∗ (x − x0 ))(e − (x − x0 ))).
where σφε (e ∗ (x − x0 ))(e − (x − x0 )) stretches along Γ to increase the
Lipschitz constant.
Setting R = diam(C ), if b − a is small enough, we obtain
g ∈M
d(f , g ) ≤ Rε
p
B(g , αRε) ∩ Na,b
= ∅.
p
is porous.
Hence Na,b
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
8 / 12
Sketch of the proof, part II
C
Γ
0
x0
Fix α > 0, ε > 0. Now set
g (x) = f (x + σφε (e ∗ (x − x0 ))(e − (x − x0 ))).
where σφε (e ∗ (x − x0 ))(e − (x − x0 )) stretches along Γ to increase the
Lipschitz constant.
Setting R = diam(C ), if b − a is small enough, we obtain
g ∈M
d(f , g ) ≤ Rε
p
B(g , αRε) ∩ Na,b
= ∅.
p
is porous.
Hence Na,b
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
8 / 12
Sketch of the proof, part III
Additionally to the condition that b − a has to be small enough, it has to
be big enough so that we can cover the whole interval (0, 1). Writing

 

[ p
[ p
N =  Nak,p ,bk,p  ∪  Na0 ,b0  ∪ N0 .
k,p
k,p
k,p
k,p
0 )
0
for suitable sequences (ak,p )k,p , (bk,p )k,p , (ak,p
k,p and (bk,p )k,p and
N0 = {f ∈ M : f |Γ = const.}
finishes the proof.
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
9 / 12
Sketch of the proof, part III
Additionally to the condition that b − a has to be small enough, it has to
be big enough so that we can cover the whole interval (0, 1). Writing

 

[ p
[ p
N =  Nak,p ,bk,p  ∪  Na0 ,b0  ∪ N0 .
k,p
k,p
k,p
k,p
0 )
0
for suitable sequences (ak,p )k,p , (bk,p )k,p , (ak,p
k,p and (bk,p )k,p and
N0 = {f ∈ M : f |Γ = const.}
finishes the proof.
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
9 / 12
The case of separable Banach spaces
If X is a separable Banach space we get the following stronger result:
Theorem (B. and Dymond, 2015)
Let X be a separable Banach space. Then there exists a σ-porous set
e ⊂ M such that for every f ∈ M \ N
e , the set
N
R(f ) = {x ∈ C :
Lip(f , x) = 1}
is a residual subset of C .
Put differently, this Theorem says that outside of a negligible set, all
mappings in the space M have the maximal possible Lipschitz constant 1
at typical points of their domain C .
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
10 / 12
The case of separable Banach spaces
If X is a separable Banach space we get the following stronger result:
Theorem (B. and Dymond, 2015)
Let X be a separable Banach space. Then there exists a σ-porous set
e ⊂ M such that for every f ∈ M \ N
e , the set
N
R(f ) = {x ∈ C :
Lip(f , x) = 1}
is a residual subset of C .
Put differently, this Theorem says that outside of a negligible set, all
mappings in the space M have the maximal possible Lipschitz constant 1
at typical points of their domain C .
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
10 / 12
Outlook
Denote by gε the function
gε : C → C , x 7→ f (x + σφε (e ∗ (x − x0 ))(e − (x − x0 ))).
The curve
[0, ε0 ) → C(X ; X ), ε 7→ gε
is Lipschitz.
Question
Can such a curve be chosen differentiable, to get information on the
directions from which the midpoints gε are approaching f ?
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
11 / 12
Outlook
Denote by gε the function
gε : C → C , x 7→ f (x + σφε (e ∗ (x − x0 ))(e − (x − x0 ))).
The curve
[0, ε0 ) → C(X ; X ), ε 7→ gε
is Lipschitz.
Question
Can such a curve be chosen differentiable, to get information on the
directions from which the midpoints gε are approaching f ?
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
11 / 12
Outlook
Denote by gε the function
gε : C → C , x 7→ f (x + σφε (e ∗ (x − x0 ))(e − (x − x0 ))).
The curve
[0, ε0 ) → C(X ; X ), ε 7→ gε
is Lipschitz.
Question
Can such a curve be chosen differentiable, to get information on the
directions from which the midpoints gε are approaching f ?
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
11 / 12
References
F.S. De Blasi and J. Myjak.
Sur la porosité de l’ensemble des contractions sans point fixe.
C. R. Acad. Sci. Paris Sér. I Math., 308(2):51–54, 1989.
C. Bargetz and M. Dymond.
σ-Porosity of the set of strict contractions in a space of non-expansive
mappings.
Preprint, 2015. http://arxiv.org/abs/1505.07656.
Christian Bargetz (University of Innsbruck)
σ-Porosity of the set of strict contractions
11 June 2015
12 / 12