Exemples : enveloppes convexes, cônes Points extrémaux d`un

K x, y ∈K⇒1
2(x+y)∈K
K
K K
K L conv(K∪L) = {(1−α)x+αy |(x, y, α)∈K×L×[0,1]}
K1, . . . , Kn
conv(K1∪. . .∪Kn) = (α1x1+. . . +αnxn| ∀i∈ {1, . . . , n}, xi∈Ki, αi≥0
n
X
i=1
αi= 1).
(Kn)n∈NKn⊆Kn+1
K=Sn∈NKn
C E
∀x∈C, ∀λ≥0, λx ∈C
X E x0∈X
Xx0={v∈E| ∃t > 0,[x0, x0+tv]⊆X}.
A x0∈A Ax0E
X⊆E x0∈X Xx0=St≥01
t(X−x0)
K x0∈K Kx0
K x0
X E
cone(X) = \
C⊆E
X⊆C
C.
cone(X)X
cone(X) = {α1x1+. . . +αkxk|k≥1, xi∈X, αi≥0}.
x K
K@(y, z, α)∈K×
K×]0,1[ (1 −α)y+αz =x. K ext(K)
A K = conv(A)
K A x ∈K\A, x 6∈ ext(K)

K∀x6=
y∈K, ∀α∈]0,1[,(1 −α)x+αy ∈int(K).
(E, k.kE)
B E
⇐⇒ ⇐⇒ ⇐⇒
B
∀x6=y∈B, ∀α∈]0,1[,k(1 −α)x+αyk<1
∀x6=y∈E, ∀α∈]0,1[,k(1 −α)x+αyk<max(kxk,kyk)
E B
u∈Rdkuk1=P1≤i≤d|ui| kuk∞= max1≤i≤d|ui|
Bk.k∞
x∈B−1< xi<1x
δ= 1 − |xi|y±=x±δeiy+6=y−∈B1
2(y−+y+) = x x
ext(B)⊆ {−1,1}dxi= (1−α)yi+αzi
α∈(0,1) yi≤xi≤zizi≤xi≤yi|xi|,|zi| ≤ 1|xi|= 1
yi=zi=xixext(B) = {(±1,...,±1)}
Bk.k1
x∈∂B xi6∈ {−1,0,+1}x
j xj∈ {±1}Pi|xi|= 1
j6=i xj6∈ {−1,0,1}δ= min(|xi|,1− |xi|,|xj|,1− |xj|)y±=x±δei∓δej
y−,i, y+,i xi|y±,i|=|xi| ± δ
|y±,j |=|xj| ∓ δky±k1=kxk1≤1x=1
2(y−+y+)
xext(B)⊆ {−1,0,1}d∩B=: Z Z =
{±ei}ei= (1 −α)y+αz α ∈(0,1)
1 = (1 −α)yi+αzi|zi|>1|yi|>1z6∈ B y 6∈ B
ext(B) = {±ei}
C0([0,1]) kfk∞= max[0,1] |f|L1([0,1])
kfk1=R1
0|f|K= conv(ext(K))
BC0([0,1])
f: [0,1] →Rx0∈[0,1]
|f(x0)|<1f B δ = 1 − |f(x0)|
ε > 0∀x∈[x0−ε, x0+ε],|f(x)|<1−δ/2
χ: [0,1] →R+x0χ(x0) = 1
χ([0,1] \[x0−ε, x0+ε] = 0
f±=f±δ
2χkf±k∞≤1f=1 2(f−+f+)
fext(B)⊆ {−1,1}[0,1] ∩B
±1
x7→ 1, x 7→ −1 ext(B) = {x7→
1, x 7→ −1}
BL1([0,1])

f∈L1([0,1]) R[0,1] |f|= 1
x0∈(0,1) R[0,x0]|f|=1
2f+= 2fχ[0,x0]f−= 2χ[x0,1]
B f =1
2(f++f−)f
ext(B) = ∅
Mn(R)nSn={S∈Mn(R)|tS=S}
S+
n={S∈ Sn| ∀x∈Rn,hx|Sxi ≥ 0}.
S∈ Sn
λ1(S)≤. . . ≤λn(S)
K={M∈ S+
n|tr(M)=1}
KSn
S∈K
Sx:= xtx x Rn
SxK
Sx= (1 −α)S0+αS1S0, S1∈K α ∈]0,1[
λn(S0) = λn(S1) = 1 1 S0=S1=Sx
K= conv(ext(K))