isoperimetres en toute simplicite.

n
p n
p
ΠLn
[A1A2],[A2A3], ...., [AnA1]
GnGm⊂ Gnm≤n
Gn\ Gm
F
F
E(F)F F
F
F
F F F.
F[0,1]
Georges LION [email protected]

a= inf
A∈F (xA) = inf
M∈E(F)(xM).
A E(F)xA=ad yAz A F
∆x=a∆∩E(F)
E(F)U V F
d
yAz A FD E F
[AD] [AE]A
F
FAi1, Ai2, ......., Aim1≤i1< i2< ..... < im≤n
E(F)Ai1, Ai2, ......., Aim
Π
Π
E({A1, ..., An}) = E(L) = E(Π)
E(Π) Π Π
a(Π) < a(E(Π))
p(Π) = A1A2+A2A3+....... +An−1An+AnA1> Ai1Ai2+........ +AimAi1=p(E(Π)).
>1E(Π)
Π Π0
Π0
Π
A, B, C AB 6=BC
B(AC)B0B0A=B0C
A(BB0)AB0+B0C < AB +BC B B0
Π Π
A, B, C B, C, D Π
BC =CD B [A, C]

A, B, C, D AB =BC =CD = 1 AD =a
0<a<3a= 1, ABCD
θ=π−d BAD θ0, θ1θ
ϕ=d BCD ϕ0ϕ1ϕ θ =θ0θ1
a
ϕ0> θ0, ϕ1< θ1.
A, B, C, D ϕ > 0ϕ=π θ =θ0
ABCD
θ ϕ
cos ϕ=−acos θ+1−a2
2.
BD21 + a2+ 2acos θ=BD2= 2 −2 cos ϕ
ϕ θ [θ0, θ1] ]θ0, θ1[ sin ϕ6= 0
dϕ
dθ =−asin θ sin ϕ
S(θ)ABCD
S(θ) = 1
2(asin θ+ sin ϕ)dS dθ =asin(ϕ−θ) 2 sin ϕθ∈]θ0, θ1[.
S(θcos θ=1−a
2.
ϕ−θ0<sin ϕ≤1
lim
θ→θ0
S0(θ)>0,lim
θ→θ1
S0(θ)<0.
S(θ)
S(θ) ]θ0, θ1[S
S0θ=ϕ θ
ABCD d ABC =d BCD.
θ=ϕ ABCD AB CD

n
Πn a(Π) ≤(p(Π))2
4ntan π
n
.
Π Π0
Π0Π Π Π0
2nsin π
n
nsin π
ncos π n
p > 0p
Π0p
p
Π0
p A1D
A1pD
K Dn
A1D
1≤i < n k det(−−−−→ AiAi+1,−−−−−→ Ai+1Ak)≥0,
kdet(−−−→
AnA1,−−−→
A1Ak)≥0,
A1A2+...... +AnA1=p
AkM n
E({A1, ...., An})D
K
Dn(A1, ...., An)∈ K E({A1, ...., An})
A(A1, ...., An) = 1
2"i=n−1
X
i=2
det(−−−→ A1Ai,−−−−−→ A1Ai+1)#.
A K (A1, A0
2, ......., A0
n)
p

n
A, B, C, D
[BC]
−→
F1[BC]
−→
F2B(AB)
−→
F3C(CD)
J[BC]
J J
−→
F2−→ F3(BC) [BC]
d
ABC =d DCB