  In this article, we use the second intrinsic volume to define a metric on the
space of homothetic classes of Gaussian bounded convex bodies in a separable
real Hilbert space. Using kernels of hyperbolic type, we can deduce that this
space is isometrically embedded into an infinite-dimensional real hyperbolic
space. Applying Malliavin calculus, it is possible to adapt integral geometry
for convex bodies in infinite dimensions. Moreover, we give a new formula for
computing second intrinsic volumes of convex bodies and offer a description of
the completion for the hyperbolic embedding of Gaussian bounded convex bodies
with dimension at least two and thus answer a question asked by Debin and
Fillastre [DF22].
