Convex Geometry of Orbits - MSRI

70 ALEXANDER BARVINOK AND GRIGORIY BLEKHERMAN
Proof. We have
dk ≤
m�
k�
i=1 j=1
n + j − i
k − j + 1 ≤
� k� �m n + j − 1
=
k − j + 1
j=1
� n + k − 1
n − 1
� m
.
Hence
� �
n + k − 1
n + k − 1 n + k − 1
lndk ≤ mln
≤ m(n − 1)ln + mk ln
n − 1
n − 1 k
� � � �
n + k − 1
k
≤ m(n − 1) ln + 1 = m(n − 1) ln + 2 ;
n − 1 n − 1
compare the proof of Corollary 3.2.
If m ≥ 3 then lnm ≥ 1 and k/(n − 1) ≥ mlnm. Since the function ρ −1 lnρ
is decreasing for ρ ≥ e, substituting ρ = k/(n − 1), we get
ρ −1 lnρ =
Therefore, for m ≥ 3, we have
If m ≤ 2 then
n − 1
k
ln
k lnm + ln lnm
≤
n − 1 mlnm .
1
2k lndk
lnm + lnlnm
≤ +
2lnm
1 1 1 1
≤ + +
lnm 2 2e ln 3 .
n − 1
k
ln
k
n − 1 ≤ e−1 ,
since the maximum of ρ −1 lnρ for is attained at ρ = e. Therefore,
1
2k lndk ≤ e −1 + 1 < 1 1 1
+ +
2 2e ln 3
The proof now follows. ˜
To understand the convex geometry of an orbit, we would like to compute the
maximum value of a “typical” linear functional on the orbit. Theorem 3.1 allows
us to replace the maximum value by an L p norm. To estimate the average value
of an L p norm, we use the following simple computation.
Lemma 3.5. Let G be a compact group acting in a d-dimensional real vector space
V endowed with a G-invariant scalar product 〈 ·,·〉 and let v ∈ V be a point.
Let Sd−1 ⊂ V be the unit sphere endowed with the Haar probability measure dc.
Then, for every positive integer k, we have
� �� �1/2k S d−1
〈c,gv〉
G
2k dg
Proof. Applying Hölder’s inequality, we get
� �� �1/2k ��
dc ≤
S d−1
〈c,gv〉
G
2k dg
�
2k〈v,v〉
dc ≤ .
d
S d−1
�
G
〈c,gv〉 2k �1/2k dg dc
.