  Intrinsic volumes of convex sets are natural geometric quantities that also
play important roles in applications, such as linear inverse problems with
convex constraints, and constrained statistical inference. It is a well-known
fact that, given a closed convex cone $C\subset \mathbb{R}^d$, its conic
intrinsic volumes determine a probability measure on the finite set
$\{0,1,...d\}$, customarily denoted by $\mathcal{L}(V_C)$. The aim of the
present paper is to provide a Berry-Esseen bound for the normal approximation
of ${\cal L}(V_C)$, implying a general quantitative central limit theorem (CLT)
for sequences of (correctly normalised) discrete probability measures of the
type $\mathcal{L}(V_{C_n})$, $n\geq 1$. This bound shows that, in the
high-dimensional limit, most conic intrinsic volumes encountered in
applications can be approximated by a suitable Gaussian distribution. Our
approach is based on a variety of techniques, namely: (1) Steiner formulae for
closed convex cones, (2) Stein's method and second order Poincar\'e inequality,
(3) concentration estimates, and (4) Fourier analysis. Our results explicitly
connect the sharp phase transitions, observed in many regularised linear
inverse problems with convex constraints, with the asymptotic Gaussian
fluctuations of the intrinsic volumes of the associated descent cones. In
particular, our findings complete and further illuminate the recent
breakthrough discoveries by Amelunxen, Lotz, McCoy and Tropp (2014) and McCoy
and Tropp (2014) about the concentration of conic intrinsic volumes and its
connection with threshold phenomena. As an additional outgrowth of our work we
develop total variation bounds for normal approximations of the lengths of
projections of Gaussian vectors on closed convex sets.
