  A formula due to Sudakov relates the first intrinsic volume of a convex set
in a Hilbert space to the maximum of the isonormal Gaussian process over this
set. Using this formula we compute the first intrinsic volumes of
infinite-dimensional convex compact sets including unit balls with respect to
Sobolev-type seminorms and ellipsoids in the Hilbert space. We relate the
distribution of the random one-dimensional projections of these sets to the
distributions $S_1,S_2,C_1,C_2$ studied by Biane, Pitman, Yor [Bull.\ AMS 38
(2001)]. We show that the $k$-th intrinsic volume of the set of all functions
on $[0,1]$ which have Lipschitz constant bounded by $1$ and which vanish at $0$
(respectively, which have vanishing integral) is given by $$ V_k =
\frac{\pi^{k/2}}{\Gamma\left(\frac 32 k +1 \right)}, \text{ respectively } V_k
= \frac{\pi^{(k+1)/2}}{2\Gamma\left(\frac 32 k +\frac 32\right)}. $$ This is
related to the results of Gao and Vitale [Discrete Comput. Geom. 26 (2001),
Elect. Comm. Probab. 8 (2003)] who considered a similar question for functions
with a restriction on the total variation instead of the Lipschitz constant.
Using the results of Gao and Vitale we give a new proof of the formula for the
expected volume of the convex hull of the $d$-dimensional Brownian motion which
is due to Eldan [Elect. J. Probab., to appear]. Additionally, we prove an
analogue of Eldan's result for the Brownian bridge. Similarly, we show that the
results on the intrinsic volumes of the Lipschitz balls can be translated into
formulae for the expected volumes of zonoids (Aumann integrals) generated by
the Brownian motion and the Brownian bridge. Also, these results have discrete
versions for Gaussian random walks and bridges. Our proofs exploit Sudakov's
and Tsirelson's theorems which establish a connection between the intrinsic
volumes and the isonormal Gaussian process.
