  The exact common information between a set of random variables $X_1,...,X_n$
is defined as the minimum entropy of a shared random variable that allows for
the exact distributive simulation of $X_1,...,X_n$. It has been established
that, in certain instances, infinite entropy is required to achieve
distributive simulation, suggesting that continuous random variables may be
needed in such scenarios. However, to date, there is no established metric to
characterize such cases. In this paper, we propose the concept of Common
Information Dimension (CID) with respect to a given class of functions
$\mathcal{F}$, defined as the minimum dimension of a random variable $W$
required to distributively simulate a set of random variables $X_1,...,X_n$,
such that $W$ can be expressed as a function of $X_1,\cdots,X_n$ using a member
of $\mathcal{F}$. Our main contributions include the computation of the common
information dimension for jointly Gaussian random vectors in a closed form,
with $\mathcal{F}$ being the linear functions class.
