This is a nearly identical question to Euclidean volume of the unit ball of matrices under the matrix norm except in the symmetric case.
Let $\mathrm{Sym}_{n \times n}(\mathbb{R})$ be the space of real-valued $n \times n$ symmetric matrices. Let $\phi : \mathbb{R}^{n(n+1)/2} \mapsto \mathrm{Sym}_{n \times n}(\mathbb{R})$ embed $\mathbb{R}^{n(n+1)/2}$ into $\mathrm{Sym}_{n \times n}(\mathbb{R})$. Consider the set $H_n = \{ v \in \mathbb{R}^{n(n+1)/2} : \| \phi(v) \| \leq 1 \}$, where $\|M\| = \max_{\|x\|=1} \|Mx\|$ is the $\ell_2 \mapsto \ell_2$ operator norm.
What is the formula for $\mathrm{Vol}(H_n)$, where $\mathrm{Vol}(\cdot)$ is the Lebesgue measure on $\mathbb{R}^{n(n+1)/2}$?
