Le Problème de Minkowski et sa généralisation

k

⊂Rn
{ei}i=1,...,r {ai}i=1,...,r
∂S ⊂P
∀FjFjFjei
PFjkeiA(Fj) = ai
k=n−1
u1, ..., uK∈RnRnα1, ..., αK∈Rn
+
P ⊂ Rn
uiαi⇐⇒ α1u1+... +
αKuK= 0
P
α1u1+... +αKuK= 0
ui
Rn
n= 4 k= 2
⊂Rn
{ei}i=1,...,r {ai}i=1,...,r
∀ > 0

∂S ⊂P
PFjkeiA(Fj) = ai
ei
S
V k ∈N
k τ :Vk→R
ΛkV k
Λ0V=RΛ1V=V
Λ∗VL∞
k=0 ΛkV
Λ∗V∧
x∈ΛkV y ∈ΛlV=⇒x∧y= (−1)kl y∧x
x∧x= 0 ∀x∈V
(x1, ..., xn)∈Vn⇐⇒ x1∧... ∧xn= 0
v∈ΛkV k k
:⇔ ∃ x1, ..., xk∈V= Λ1V, v =x1∧... ∧xk
k V Λk
sV
k ξ ⇐⇒ Tξ:= {v∈V|v∧ξ= 0}
k
ξ η Tξ=Tη⇐⇒ ξ=cη c ∈R
ξ∈ΛkV η ∈ΛlV=⇒(ξ∧η6= 0 ⇔
Tξ∧η=Tξ⊕Tη)
ξ∈ΛkV=⇒(Tξ⊂Tη, η ∈ΛlV
⇔ ∃ ζ∈Λl−kV η =ξ∧ζ)

Grk(V)
k V Grk(V)=Λk
sV/ ∼
ξ∼η:⇔ξ=cη c ∈Rξ, η
Gr+
k(V)k
V
V=Rn
k k
k G(k, n)
GC(k, n)kRn
M k ∈N
ω:M−→ ΛkT∗M=Sp∈MΛkT∗
pM k
:⇐⇒ prp◦ω=idMprpΛkT∗M→M
p∈M, ω(p)∈ΛkT∗
pM
k M Ωk(M)
M n ω
ω=P1≤i1≤...≤ik≤nfi1,...,ikdxi1∧... ∧dxikω=PIfIdxI
I={i1, ..., ik}k < n
Φ : M−→ N
M N ω N
Φ∗ωΦ∗ω=PI(fI◦Φ)dΦIdΦI
ΦM
(k+ 1)
k
Ωk(M)−→ Ωk+1(M)
f∈Ω0(M)f M df
f
R−