137 - Barycentres dans un espace affine réel de dimension finie

E F n m
K E F
A1, . . . , Ak∈E λ1,...λk∈KPλ16= 0
Pλi
−−→
GAi=~ 0
∀M∈ E,Pλi
−−→
MG =Pλ−−−→ MAi
G
(Ai, λi)Bar((Ai, λi),1≤
i≤k)
Bar((e2ikπ/n,1),0≤k < n)=0
∀µ6= 0, Bar((Ai, λi)) = Bar((Ai, µλi))
(Ai,j)1≤j≤ki,1≤i≤k
(λi,j)1≤j≤ki,1≤i≤k∀i, Pki
j=1 λi,j 6= 0
i Gi=Bar((Ai,j, λi,j),1≤j≤kj)G=Bar((Ai,j , λi,j),1≤
j≤kj,1≤i≤k)G=Bar((Gi,Pki
j=1 λi,j),1≤i≤k)
f:E → F f
car(K) =6= 2

X⊂ E X6=∅X
hXiX
hXiX
(A1, . . . , Ak)∈ Ek(A1, . . . , Ak)
M Ai
∃!(λ1,...λk)∈Kk:M=Bar((Ai, λi)) Pλi= 1
(λ1, . . . , λk)∈Kk
M=Bar((Ai, λi)) M∈ E
{Ai}
1/3,1/3,1/3
(A0, . . . , Ak)∈ EkE(−−−→ A0A1,...,−−−→ A0Ak)
E
n+ 1
(A0, . . . , An)EB0, . . . , Bm
Ff:E → F f(Ai) =
Bi∀i f (B0, . . . , Bm)F
A1A2A3
(λ1, λ2, λ3)
σ∈S3Gσ=Bar((Ai, λσi)) Gσσ∈S3
A1A2A3
(A0, . . . , An)EH
Kn+1 Pxi= 1 E → H, A =Bar((Ai, λi)) 7→
(λ0, . . . , λn)
Kn+1
Kn
(P0, . . . , Pn)∈ En
(A0A1A2)
A=Bar((Ai, λi)) B=Bar((Ai, βi)) B6=A M =
Bar((Ai, xi)) (AB)
x0α0β0
x1α1β1
x2α2β2
= 0

K=R
C⊂ E
car 6= 2
Gln(R)
X⊂ E, X 6=∅X
co(X)X
X
co(X)
X
f:E→Rf epi(f) = {(x, t)∈E×R|
f(x)≤t}
X⊂ E C=co(X)C
n+ 1 X
X⊂ E X6=∅A /∈X D =d(A, X)
X d
X d
X⊂ E X6=∅ ∀A∈ E,∃!p(A)∈X:
d(A, p(A)) = d(A, X)X
C⊂E A ∈C A
C A =Bar((B1, λ1),(B2λ2)) B1, B2∈C
B1=B−2
A C \ {A}
C
K
C f :E → E
f(C) = C f C

Gln
KRn
f:K→K
ABC ABC =
Fa∪Fb∪FcFa, Fb, Fc[A, B],[B, C]
[C, A]
H
A B H
A B f ∈H∗
supa∈Af(a)<infb∈Bf(b)