Let $E$ be the real vector space of $n\times n$ real symmetric (resp. complex Hermitian) matrices, and $E_1$ those with trace $1$. Endow $E$ with the bilinear (resp. sesquilinear) form given by $(P,Q) \mapsto \frac{1}{n}\operatorname{tr}(P^* Q)$ (taking the convention that Hermitian forms are linear on the right and antilinear on the left, and where $P^*$ refers to the conjugate transpose of $P$), and $E_1$ with the Euclidean structure this induces.
Inside $E_1$, we have the compact convex set $\mathcal{S}$ of positive semidefinite matrices with trace $1$. This set is sometimes known as the “spectraplex” (in the context of semidefinite programming) or, for the complex case, “Bloch body” (in quantum mechanics, where its elements are known as “density matrices” or “mixed states”).
QUESTION: What is the $\frac{1}{2}(n^2+n-2)$-dimensional (resp. $(n^2-1)$-dimensional) volume of $\mathcal{S}$? What is its boundary surface measure? What about lower-dimensional intrinsic volumes (≈ quermaß measures)?
Remarks: Given the importance of $\mathcal{S}$ in many contexts such as positive semidefinite programming, quantum mechanics / quantum information theory, etc., I find it hard to believe that its volume is not a well-known quantity (or at least “famously unknown” like the Birkhoff polytope); but as there are so many different terms for everything involved (and also two different cases, real and complex), I may well have missed the right terms to search.
Concerning the boundary $\partial\mathcal{S}$, it is worth noting that an important part of it is given by the rank $1$ matrices (i.e., orthogonal projectors on $1$-dimensional subspaces), which is just a projective space. More precisely, putting $\mathbb{K} = \mathbb{R}$ (resp. $\mathbb{K} = \mathbb{C}$) and endowing $\mathbb{K}^n$ with the standard bilinear (resp. sesquilinear) form given by $(u,v) \mapsto u^* v$ (seeing $u,v$ as column vectors), then the (“Veronese”) map $v \mapsto v v^*$ from $\mathbb{K}^n$ to $E$ takes the norm-$1$ sphere in $\mathbb{K}^n$ to the subset $\mathcal{P}$ of $E_1$ consisting of rank $1$ matrices (physicists call them “pure states”); obviously this map factors out by the scalars of modulus $1$, so it defines a bijection $\mathbb{P}^{n-1}(\mathbb{K}) \to \mathcal{P}$. But in fact, this is easily seen to be (up to a constant $\sqrt{2}$ factor) an isometry of $\mathbb{P}^{n-1}(\mathbb{K})$ (endowed with its standard Fubini-Study metric) to $\mathcal{P}$ (seen as a submanifold of $E_1$). Now the volume of $\mathbb{P}^{n-1}(\mathbb{K})$ is known, so the measure of $\mathcal{P}$ also is. But while $\mathcal{P}$ is the set of extremal points of $\mathcal{S}$ (by the spectral theorem, any positive definite matrix is a convex combination of projectors), it is of lower dimension than $\partial\mathcal{S}$ when $n\geq 3$.
