Let $\Bbb{R}^n$ be the Euclidean space of $n$-dimensional column vectors with real coefficients. Moreover, $\Bbb{S}_{++}^n$ be the space of symmetric positive definite $n\times n$ real matrices. We construct a new space $\mathcal{X}=\Bbb{R}^n\times\Bbb{S}_{++}^n$, whose members are of the form $(\mathbf{x},A)$, where $\mathbf{x}\in\Bbb{R}^n$ and $A\in\Bbb{S}_{++}^n$. What kind of space is this? Could we have an intuition on it's "shape"? Could we define some norm in this space (via some inner product)? Also, could $\Bbb{R}^n$ be seen as a projection of $\mathcal{X}$?
Finally, if we define a function $f\colon\mathcal{X}\to\Bbb{R}$, what would be the following: $f(\mathbf{x})$ for some constant $A\in\Bbb{S}_{++}^n$?
Please feel free to correct me wherever you find useful. Thanks a lot!
Since $\mathbb R^n$ is convex (affine), and $\mathbb S^n_{++}$ is a convex cone, the product $\mathbb R^n\times\mathbb S^n_{++}$ is also a convex cone. This would be the shape you are looking for. For $n=1$, this is just the open right half plane.
$\mathbb S^n_{++}$ is also a $\frac{n(n+1)}{2}$ dimensional manifold with tangent space at a point $P\in\mathbb S^n_{++}$ given by $\{p\}\times\mathbb S^n$ where $\mathbb S^n$ is the collection of all symmetric $n\times n$ matrices. A Riemannian metric can be given as follows: $$ \langle A,B\rangle_P:=\text{tr}(P^{-1}AP^{-1}B),\quad A,B\in\mathbb S^n $$ So $\mathbb S^n_{++}$ and therefore $\mathcal X:=\mathbb R^n\times \mathbb S^n_{++}$ is also a Riemannian manifold.
A good summary of three aspects of $\mathbb S^n_{++}$, namely, convex cone, Riemannian manifold, and Jordan algebra can be found in this paper.