Hyperbolic embedding of infinite-dimensional convex bodies

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].