which implies that P
∂ D
j
∋x
c
i
n
i
x belongs to the set N
x,
α
of normal vectors on the boundary of D,
for α = β
min
∂ D
j
∋x
α
j
. Thanks to remark 3.5, this proves the inclusion
N
′
x
:=
n ∈ S
m
, n =
X
∂ D
i
∋x
c
i
n
i
x with c
i
≥ 0
⊂ N
x, α
Let us prove the converse inclusion. For i such that ∂ D
i
∋ x, the boundary ∂ D
i
is at least C
2
in
D, hence there exist a closed ball Bx, η
x i
with η
x i
0, and a C
2
function f
x
i
: B x,
η
x i
−→ R with non-vanishing derivative, such that
B x,
η
x i
∩ D
i
= Bx, η
x i
∩ {y ∈ R
m
, f
x
i
y ≥ 0}
there even exists an orthonormal coordinate system in which f
x
i
y is the difference between the
last coordinate of y and a
C
2
function of the other coordinates, but we do not need this precise form here. The Taylor-Lagrange formula provides :
∀z s.t. |z| ≤ η
x i
f
x
i
x + z = f
x
i
x + ∇ f
x
i
x.z +
1 2
z.D
2
f
x
i
z
∗
z
for some z
∗
∈ Bx, η
x i
depending on z. By definition of f
x
i
, x
∈ ∂ D
i
implies f
x
i
x = 0 and ∇ f
x
i
x =
|∇ f
x
i
x|n
i
x. Moreover, the second derivative D
2
f
x
i
is continuous hence bounded on the closed ball by some constant
||D
2
f
x
i
||
∞
:
∀z s.t. |z| ≤ η
x i
f
x
i
x + z ≥ |∇ f
x
i
x|n
i
x.z −
||D
2
f
x
i
||
∞
2
|z|
2
For any ǫ 0 and for δ
i ǫ
x = min
η
x i
, 2
|∇ f
x
i
x|
||D
2
f
x
i
||
∞
ǫ
we obtain :
|z| ≤ δ
i ǫ
x and z.n
i
x ≥ ǫ|z| =⇒
f
x
i
x + z ≥ 0 =⇒
x + z ∈ D
i
Consequently, for N
ǫ
= \
∂ D
i
∋x
{z, n
i
x.z ≥ ǫ|z|} we have : {x + z, z ∈ N
ǫ
and |z| ≤ min
i
δ
i ǫ
} ⊂ D By definition, for each
n
∈ N
D
x
, there exist α
n
0 such that ∀y ∈ D y − x.n +
1 2
α
n
|y − x|
2
≥ 0, hence for
z ∈ N
ǫ
and λ 0 small enough :
λz.n +
λ
2
2 α
n
|z|
2
≥ 0 For this to hold even with
λ going to zero, z.n has to be non-negative. So we obtain : ∀n ∈ N
D
x
∀ǫ 0 − n ∈ N
∗ ǫ
where N
∗ ǫ
= {v, ∀z ∈ N
ǫ
v.z
≤ 0} is the dual cone of the convex cone N
ǫ
. As proved in Fenchel [3] see also [9], the dual of a finite intersection of convex cones is the set of all limits of linear
combinations of their dual cones, in particular : N
∗ ǫ
= X
∂ D
i
∋x
{z, n
i
x.z ≥ ǫ|z|}
∗
= X
∂ D
i
∋x
{v, − n
i
x.v ≥
p 1
− ǫ
2
|v|}
152
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
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