In order to estimate the size of the torsion in the algebraic $K$-groups of $\mathbb Q$ one needs to understand the homology of $\mathrm{GL}_n(\mathbb Z)$, or alternatively, the homology of the space of so-called "well rounded" lattices. This is difficult. While trying to get some rough estimates, the following counting elementary problem came up.

Let $n\geq 1$ be an integer, let $r>0$ be real, and write $B \subseteq \mathbb Z^n$ for the set of integer vectors of length $\leq r$. Count the subsets $C\subseteq B$ with the following properties:

(1) $C = -C$ (central symmetric) and $0\in C$

(2) The only integer vector in the relative interior of the convex hull of $C$ is $0$.

I am equally interested in counting these subsets up to the action of the symmetric group $\mathfrak S_n$ acting by permuting coordinates.

Is there any literature on this kind of problem? What keywords should I search for?


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