  Given n (discrete or continuous) random variables X_i, the
(2^n-1)-dimensional vector obtained by evaluating the joint entropy of all
non-empty subsets of {X_1,...,X_n} is called an entropic vector. Determining
the region of entropic vectors is an important open problem with many
applications in information theory. Recently, it has been shown that the
entropy regions for discrete and continuous random variables, though different,
can be determined from one another. An important class of continuous random
variables are those that are vector-valued and jointly Gaussian. In this paper
we give a full characterization of the convex cone of the entropy region of
three jointly Gaussian vector-valued random variables and prove that it is the
same as the convex cone of three scalar-valued Gaussian random variables and
further that it yields the entire entropy region of 3 arbitrary random
variables. We further determine the actual entropy region of 3 vector-valued
jointly Gaussian random variables through a conjecture. For n>=4 number of
random variables, we point out a set of 2^n-1-n(n+1)/2 minimal necessary and
sufficient conditions that 2^n-1 numbers must satisfy in order to correspond to
the entropy vector of n scalar jointly Gaussian random variables. This improves
on a result of Holtz and Sturmfels which gave a nonminimal set of conditions.
These constraints are related to Cayley's hyperdeterminant and hence with an
eye towards characterizing the entropy region of jointly Gaussian random
variables, we also present some new results in this area. We obtain a new
(determinant) formula for the 2*2*2 hyperdeterminant and we also give a new
(transparent) proof of the fact that the principal minors of an n*n symmetric
matrix satisfy the 2*2*...*2 (up to n times) hyperdeterminant relations.
