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

SIZES OF COMPACT SUBSETS OF HILBERT SPACE 329
of C, (Bonnesen and Fenchel [.?I, Paragraph 29, p. 38; Paragraph 32,
p. 49; Paragraph 38, p. 61). I believe some estimates of Santalb [16]
are of this sort.
However, such estimates do not seem adequate for our purposes.
Consider for example Oc ({log n)-‘l”)). To estimate N(Oc, E) is more
or less equivalent to estimating N(Oc, , c/2) where Oc, is the intersection
of Oc with the span of v1 ,..., vn , and n is approximately
e4/Ea. Then every point of the boundary of Oc, is at least (n log n)-li2
from the origin, so
AJoc,q > c&y 1 + 1/2n19n,
Wk%~~ B exp (Y exp W2))
if y < Q and 12 is large enough. Thus this method seems quite inferior
to that used to prove Proposition 6.13, in this case, since it produces
an extra exponentiation.
ACKNOWLEDGMENTS
I am greatly indebted to Volker Strassen for the idea of introducing a-entropy into
the study of sample continuity of Gaussian processes, and for the statement of the
result which now appears as Corollary 3.2.
Another main result, Theorem 5.3, is proved using L.A. Santalb’s theorem [17] on
volumes of convex symmetric sets and their polars. I thank G. D. Chakerian for
telling me of Santalb’s result via a network of mutual friends.
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