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

SIZES OF COMPACT SUBSETS OF HILBERT SPACE 309
Then by Stirling’s formula there is a p > 4 such that
for n large enough, and a, < n-b. Thus 5.7 and 3.2 imply that C is a
GC-set. Q.E.D.
Suppose given a Banach ball (= convex symmetric bounded
closed set) C in H. Suppose also that {FJ is a sequence of subspaces
adapted to C. Given Fl ,..., F,-, , we assume F, can be and is chosen
among its possible values so as to minimize h,(F, n C). Then we
define
Wn = UFn n C>,
EW(C) = liT+iup (log Wn)/(n log n).
For a sufficiently “smooth” set C, e.g., an ellipsoid, we shall have
W(C) = EW(C) and even V, = W, (see Proposition 6.1 below).
At the end of Section 6 we show that EW(C) < EV(C) is possible.
Next we obtain a lower bound for r(C) in terms of EV(C). In each
of the four classes of examples treated in Section 6, it becomes an
equality at least for EV( C) < - 1.
PROPOSITION 5.8. For any convex symmetric set C in H,
(a) r(C) > - 2/(2EV(C) + 1) if EV(C) < - 4
(b) r(C) = + co &W(C) > - &.
Proof. If C is covered by m sets, each of diameter at most E, then
any n-dimensional projection P,C is covered by m balls of radius E,
and
mc,P 2 V, , so w, 4) >, Vn/w”
for all n. Let EV(C) = - b > - c and c > +. Then for n large
enough
V,/c# > nn’z(~)1/2/[(27fe)“/2 &znc] = k, ,
say. The following paragraph gives motivation only.
To maximize k, , we note that
kn+$tn = ((n + l)/n)n[(l’+cl (n + 1)(1-c)/42m)l’2,
which is asymptotic as n + co to
e-cn(1/2)-c/E(2,)1/2.