ensembles convexes en dimension finie - IMJ-PRG

E d
E
E
E=Rd
E
k · k h·,·i
C⊂E
A, B ∈C[A, B]C
∀(A, B)∈C2,∀t∈[0,1],(1 −t)A+tB ∈C.
B∞(0,1) = (x1,··· , xd)∈Rd|maxi|xi| ≤
1
(x1,··· , xd)∈Rd|maxi|xi|<1
B(0,1) = x∈E| kxk ≤ 1
C E C
C={λx |x∈C, λ ∈R+}
C(Pi)1≤i≤nC
(ai)1≤i≤n0≤ai≤1Pn
i=1 ai= 1 Pn i=1 aiPi
C
A⊂E A conv(A)
Aconv(A)
A

A E d
d+ 1 A
K⊂E
F⊂E(xn)n∈NF
y y ∈F
f:
K→RK
ER
`
ker `={x∈E|`(x) = 0}
E E dim E−1
C E x ∈E
E pC(x)
kx−pC(x)k= min
y∈Ckx−yk.
y∈C
hx−pC(x), y −pC(x)i ≤ 0.
pCkpC(x)−pC(y)k≤kx−yk
x /∈C ` E α ∈R`(x)> α
`(y)< α y ∈Ckx−pC(x)k>0
` `(y)≤`(x)
y∈C
C−x
x C r > 0B(x, r)C{

C E x ∈C C
C\ {x}
x∈C y, z ∈C
λ∈[0,1] x=λy + (1 −λ)z x =y=z
E
x6=y E ` E
`(x)< α < `(y) ker(`−α)
x y
F⊂C C x, y ∈C
λ∈]0,1[ λx + (1 −λ)y∈F x, y ∈F
`
F`=x∈C|`(x) = max
y∈C`(y)
C
C
dim E C
M∈Mn(R)
1≤i≤n
n
X
j=1
mi,j = 1
n
X
j=1
mj,i = 1.
M∈Mn(R)σ∈Sn
mi,j = 1 σ(j) = i
BMn(R)
BB
M∈ B
B
EtF
E F

C⊂E C C0
C0={y∈E| ∀x∈C, hx, yi ≤ 1}.
C0C
C0r > 0B(0, r)⊂C
C0
C C00
C⊂(C0)0
B1(0,1) = (x1,··· , xd)∈
Rd|Pi|xi| ≤ 1
B∞(0,1) =
(x1,··· , xd)∈Rd|maxi|xi| ≤ 1
CAA
E
`(x)≤α ` α
P1,··· , Pm:Rd→R
P1(x),··· , Pm(x)≥0x∈
Rd
q1,··· , qm≥0q1P1+··· +qmPm=−1
d
Pi`i+αi`iαi

CACBE
P:E→RP(x)≤ −1x∈CAP(x)≥1x∈CB
A
B CACB
CA(C0
A)0=CA
P≥1P≤ −1
AB
A B
C C ∩ {x|`(x)≤α}
` α C
k C d −1C k −1
k
k
1
C1,··· , Cn
d+ 1 d+ 1
Ci
n=d+ 2 aj∈Ti6=jCi∆k=
conv{aj|j6=k}