24
VIII – Cauchy Theory
The image σ(I × I) ⊂ G being compact, as already stated, its distance R
from the border of G is > 0. Choose r < R. As σ is uniformly continuous on
the compact set I × I, there exists r
> 0 such that
|s − s
| ≤ r
& |t − t
| ≤ r
=⇒ |σ(s, t) − σ (s
, t
)| ≤ r .
(3.10)
For t = t
, in particular this shows that
|s − s
| ≤ r
=⇒ ⇒μ s − μ s I ≤ r ;
(3.11)
see remarks at the end of (i).
Having said that, let us choose an integer n, give s values of the form
s p = p/n with 0 < p < n and let ν p be the path μ s for s = p/n. It may
not be admissible, but it can be approached by a piecewise linear, and hence
admissible, path γ p by choosing for its successive vertices the points of ν p
indexed by the parameter t q = q/n, i.e. the points a pq = σ(p/n, q/n).
0
1
1
1
1
1
1
1
μ
μ
γ
γ
γ
ν
ν
ν
Fig. 3.3.
If n is sufficiently large, the diameter of the square K pq with vertices
(s p , t q ), (s p+1 , t q ), (s p+1 , t q+1 ), (s p+1 , t q+1 ) in I×I is < r
. Therefore, by (10),
its image under σ is contained in the disc D pq centered at a pq and of radius r.
However, the latter is convex and contained in G since r < R. As a result :
(a) the line segment connecting a pq to a p,q+1 is contained in G for all q,
so that the same holds for the path γ p obtained by juxtaposing these
segments for different values of q;
(b) for any t ∈ [t q , t q+1 ], the line segment connecting σ (s p , t) to σ (s p+1 , t)
is contained in G since its endpoints are in D pq . Hence, there is a linear
deformation that takes us from γ p to γ p+1 without leaving G.
The same arguments show that there are linear deformations taking us
from μ 0 to γ 1 and from γ n−1 to μ 1 .
VIII – Cauchy Theory
The image σ(I × I) ⊂ G being compact, as already stated, its distance R
from the border of G is > 0. Choose r < R. As σ is uniformly continuous on
the compact set I × I, there exists r
> 0 such that
|s − s
| ≤ r
& |t − t
| ≤ r
=⇒ |σ(s, t) − σ (s
, t
)| ≤ r .
(3.10)
For t = t
, in particular this shows that
|s − s
| ≤ r
=⇒ ⇒μ s − μ s I ≤ r ;
(3.11)
see remarks at the end of (i).
Having said that, let us choose an integer n, give s values of the form
s p = p/n with 0 < p < n and let ν p be the path μ s for s = p/n. It may
not be admissible, but it can be approached by a piecewise linear, and hence
admissible, path γ p by choosing for its successive vertices the points of ν p
indexed by the parameter t q = q/n, i.e. the points a pq = σ(p/n, q/n).
0
1
1
1
1
1
1
1
μ
μ
γ
γ
γ
ν
ν
ν
Fig. 3.3.
If n is sufficiently large, the diameter of the square K pq with vertices
(s p , t q ), (s p+1 , t q ), (s p+1 , t q+1 ), (s p+1 , t q+1 ) in I×I is < r
. Therefore, by (10),
its image under σ is contained in the disc D pq centered at a pq and of radius r.
However, the latter is convex and contained in G since r < R. As a result :
(a) the line segment connecting a pq to a p,q+1 is contained in G for all q,
so that the same holds for the path γ p obtained by juxtaposing these
segments for different values of q;
(b) for any t ∈ [t q , t q+1 ], the line segment connecting σ (s p , t) to σ (s p+1 , t)
is contained in G since its endpoints are in D pq . Hence, there is a linear
deformation that takes us from γ p to γ p+1 without leaving G.
The same arguments show that there are linear deformations taking us
from μ 0 to γ 1 and from γ n−1 to μ 1 .