While trying to use Minkowski's theorem to calculate the (left) class number of a noncommutative ring, I ran into the following problem:



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> What is the volume of the largest symmetric convex subset $S$ of $\mbox{Mat}\_3(\mathbb{R})$ such that every matrix in this subset has determinant at most $1$? In particular, is there such a set with volume greater than 885?
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The best subset that I have been able to find so far is


$S\_1 = \{M\in\mbox{Mat}\_3(\mathbb{R})\mid \forall O\in O(3,\mathbb{R}), \mbox{ Tr}(MO) \le 3\}$.


If we apply Gram-Schmidt to the rows of $M$ and use the resulting orthonormal basis for the columns of an orthogonal matrix $O$, then by the AM-GM inequality we can conclude that $\det M \le 1$ from $\mbox{Tr}(MO)\le 3$. Furthermore, if $S$ is any symmetric convex set of matrices of determinant at most $1$ which contains an orthogonal matrix $O$, then every $M\in S$ must satisfy $\mbox{Tr}(MO^{-1}) \le 3$ by considering the tangent hyperplane to the collection of matrices with determinant $1$. Thus $S\_1$ is the largest such set which contains all orthogonal matrices.


Unfortunately, it's incredibly difficult to calculate the volume of $S\_1$. All I've been able to do so far is to note that it contains the set $S\_r$ of matrices such that the sum of the norms of the rows is at most $3$, and that the volume of $S\_r$ is $\frac{972\pi^3}{35}\approx 861.089$. If we let $S\_c$ be the similar set obtained by considering the columns, then Monte-Carlo integration indicates that the volume of $S\_r\cup S\_c$ is around $1050$, but I haven't been able to prove this.



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> What is the volume of $S\_1$? If this is too hard, what is the volume of $S\_r\cup S\_c$?
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Motivation
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The ring I am dealing with is $\mathbb{Z}\langle a,x\rangle/(a^3 = a-1, x^3 = 3x-1, xa = a(x^2-2)+2-x)$. It is an order in the central simple algebra over $\mathbb{Q}$ with invariants $1/3$ at $3$, $2/3$ at $2$, and $0$ at every other prime. The discriminant of this ring is $-6^6$ (as one can verify by working out the trace form by hand).


Since the associated central simple algebra has nontrivial invariants at $2$ and $3$, and since $\mbox{Nm}(x+1) = -3$ and $(a+xa-1)^3 = 2$, all left ideals of norm at most $4$ are principal. If we tensor this ring up to $\mathbb{R}$ it becomes $\mbox{Mat}\_3(\mathbb{R})$, so we can think of it as a lattice in $\mbox{Mat}\_3(\mathbb{R})$ of covolume $\sqrt{6^6}$. By Minkowski's bound, every left ideal class contains a representative of norm at most $\left(\frac{2^9\sqrt{6^6}}{\mbox{Vol}(S)}\right)^{1/3}$, so if we can find a set $S$ with volume at least $885$ then we can conclude that this ring has left class number $1$.



Geometry of numbers for three by three matrices?
The volume of $S\_1$ is $\frac{16767 \pi^4}{560} \approx 2916.53$.


The idea is to realize that the set $S\_1$ can be equivalently described as the nuclear norm (sum of the singular values $\sigma\_i$) of $M$ being less than or equal to 3, i.e., $S\_1 = \{M : \sum\_i \sigma\_i(M) \leq 3 \}$. 


This is an orthogonally invariant set, so from the singular value decomposition $M = U \Sigma V^T$ we can do the integral pretty explicitly, using the fact that the Jacobian is given by $(\sigma\_1^2-\sigma\_2^2) (\sigma\_1^2-\sigma\_3^2)(\sigma\_2^2-\sigma\_3^2)$, and the integral is over the polytope $\{(\sigma\_1,\sigma\_2,\sigma\_3) : \sigma\_1 \geq \sigma\_2 \geq \sigma\_3 \geq 0, \, \sigma\_1 + \sigma\_2 + \sigma\_3 \leq 3\}$ (this integral is $\frac{16767}{17920}$). The other factor corresponds to the orthogonal matrices $U,V$, and is $\frac{1}{2^n} Vol(O^n)^2= \frac{1}{2^n} (\prod\_{i=1}^n Vol(S^{i-1}))^2 = \frac{1}{2^n} (\prod\_{i=1}^n \frac{2 \pi^{i/2}}{\Gamma(i/2)})^2$, which for $n=3$ gives $32 \pi^4$.


