DévAlg 21: Ellipsoïde de John(-Loewner

S++
nnh·,·i
Rnu∈ S++
nBu={x∈Rn/hux, xi ≤ 1}
Rnqu=hu·,·i u7→ BuS++
n
Rn0
KRn
Vect (K) = Rn0K
K
kAkK= supx∈KkAxk Mn(R)
u∈ S++
nv∈ S+ ntvv =uBu=v−1Bn
BnRnVol Bu=Vol Bn
√det u
S++
nMn(R)u16=u2∈ S++ nt∈]0,1[
ln dettu1+ (1 −t)u2> t ln det u1+ (1 −t) ln det u2
CK={u∈ S++
n/Bu⊃K} CKu0∈ CK
Cu0={u∈ CK/ln det u≥ln det u0}
ln det Cu0
Gln(R)
G Gln(R)g∈Gdet g=±1
(det g)n
K=[
g∈G
g(Bn)G
EK G gE=Eg∈G
v∈ S+
nvgv−1(Bn) = Bng∈G vGv−1⊂O(n)
FRnG G