For k ∈ N large enough, since −n ∈ N
∗
1 k
, there exist unit vectors n
i,k
and non-negative numbers c
i,k
such that :
n =
X
∂ D
i
∋x
c
i,k
n
i,k
and for
∂ D
i
∋ x n
i
x.n
i,k
≥ r
1 −
1 k
2
When k tends to infinity, n
i,k
tends to n
i
x, and for k large enough n
i,k
. l
x
≥
β 2
thus : 1
≥ n.l
x
≥ X
∂ D
i
∋x
c
i,k
n
i,k
. l
x
≥ β
2 X
∂ D
i
∋x
c
i,k
Thus the sequences c
i,k k
are bounded, which implies the existence of convergent subsequences. Their limits c
i, ∞
≥ 0 satisfy :
n =
X
∂ D
i
∋x
c
i, ∞
n
i
x
This completes the proof of N
D
x
⊂ N
′
x
. We already proved that N
′
x
⊂ N
D
x, α
with α = β
min
∂ D
j
∋x
α
j
, so we obtain
N
D
x
= N
′
x
= N
D
x, α
for each x
∈ ∂ D. As a consequence, D ∈ U ESβ min
1 ≤ j≤p
α
j
. Let us now prove that
D ∈ UN Cβ, δ. The Uniform Normal Cone property restricted to D holds for D
i
, with constant β
i β
2
2
≤
1 2
. That is, for x
∈ D ∩ ∂ D
i
there exist a unit vector l
i x
which satisfies
l
i x
. n
i
y ≥
p 1
− β
2 i
for each y
∈ D ∩ ∂ D
i
such that |x − y| ≤ δ
i
. If
l
i x
= n
i
x, this implies that |n
i
x − n
i
y|
2
≤ 2β
2 i
. If
l
i x
6= n
i
x, we use the Gram-Schmidt orthogonalization process for a sequence of vectors
with
l
i x
and
n
i
x as first vectors, then compute n
i
x.n
i
y in the resulting orthonormal basis l
i x
, e
2
, e
3
, . . . , e
m
:
n
i
x.n
i
y = l
i x
. n
i
yl
i x
. n
i
x + n
i
y.e
2
Æ 1
− l
i
x
. n
i
x
2
Note that |n
i
y.e
2
| ≤ β
i
because l
i x
. n
i
y ≥
p 1
− β
2 i
and |n
i
y| = 1, thus : n
i
x.n
i
y ≥
Æ 1
− β
2 i
2
− β
2 i
= 1 − 2β
2 i
This implies that |n
i
x − n
i
y|
2
≤ 4β
2 i
. So in both cases :
|n
i
x − n
i
y| ≤ 2β
i
as soon as |x − y| ≤ δ
i
for x, y
∈ D ∩ ∂ D
i
. Let us now fix
x ∈ ∂ D and δ = min
1 ≤i≤p
δ
i
2. We then choose x
′
∈ ∂ D ∩ Bx, δ such that {i s.t. x
′
∈ ∂ D
i
} ⊃ {i s.t. y ∈ ∂ D
i
} for each y ∈ ∂ D ∩ Bx, δ and we let l = l
x
′
. To complete the proof of
D ∈ UN Cβ, δ, we only have to prove that n.l is uniformly bounded from below for n
∈ N
D
y
with y
∈ ∂ D ∩ Bx, δ.
We already know that each n
∈ N
D
y
is a convex sum of elements of the N
D
i
y
: n =
P
∂ D
i
∋y
c
i
n
i
y.
The coefficients c
i
are non-negative, their sum is not smaller than 1 because n is a unit vector, and
is not larger than
1 β
thanks to i v. So the vector n
′
= P
∂ D
i
∋y
c
i
n
i
x
′
satisfies :
n
′
. l =
X
∂ D
i
∋x
′
c
i
n
i
x
′
.l ≥ β and
|n
′
− n| ≤
X
∂ D
i
∋y
c
i
|n
i
x
′
− n
i
y| ≤ 2
X
∂ D
i
∋y
c
i
β
i
≤ 2 max
1 ≤i≤p
β
i
β Consequently :
n.l ≥ n
′
. l
− |n
′
− n| ≥ β − 2 max
1 ≤i≤p
β
i
β
0. 153
4 Existence of dynamics for globules and linear molecule models
4.1 Globules model
