On the Characters and the Plancherel Formula of Nilpotent Groups ...

306 DUDLEY
positive number leaves the relevant properties unchanged, so we may
assume M = 1. Suppose the first conclusion is false. Then for any
K > 0 there is an n such that V, > (K/n)lz.
Let P, be a projection with n-dimensional range F. Then
Y < Pr (-@‘,C) < 1) = G(P,CY),
where the polar is taken in the dual of F and G is normalized Gaussian
probability measure. We use the general inequality
where B is any convex symmetric set in Rn (due to Santa16 [17]).
(Later work by Bambah [I] on a lower bound for h,(B) h,(P) may
also be noted.) For any /3 > 0 there is a P, such that
U~nC) 2 tqy, so h&w7m G ca2(nP)n-
Using (5.1) we obtain for any 01 > 0
M~nq) < 4cw’2
for certain arbitrarily large n. Now, given X,(A) for a set A, G(A)
is clearly maximized when A is a ball E(r) centered at 0, say of radius T.
Hence
where r, < (~ln)l/~. Then
where
G(PnW < ‘Wt~,d),
G(E(r,)) < jr"" ~+le-‘~/2 dr/I, ,
I,= I m 0
~--l~=l~ dr.
The integrand increases for 0 < r < (n - 1)li2. But (an/(n - 1))li2 --t 0
as n -+ co and 0110, so G(E(r,)) -+ 0 as n + co through a suitable
sequence, contradicting the fact that G((P,C)l) >, y > 0.
Thus for some M > 0,
log vv2
M
-