<p>In this article, we use the second intrinsic volume to define a kernel of hyperbolic type on the space of homothety classes of Gaußian bounded convex bodies in a separable real Hilbert space. With this kernel, we deduce that this space can be embedded into an infinite-dimensional real hyperbolic space and is then equipped with a hyperbolic metric, which is the “area distance” introduced by Debin and Fillastre (Gr Geom Dyn 16(1):115–140, 2022). Applying the Malliavin calculus, it is possible to adapt integral geometry for convex bodies in infinite dimension. Moreover, we give a new formula for computing the second intrinsic volumes of convex bodies and a characterisation of the equality case of Alexandrov–Fenchel inequality in infinite dimension, offering 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 (2022).</p>