Can someone tell me whether this space has a name? Each element of the space is an $n$-dimensional point (tuple) (vector, but not sure it's a vector space?) where each element of the tuple may be either $-1$, $0$, or $1$?
It seems to me more or less the corners of a $n$-dimensional hyper cube plus all the midpoints of the edges and $(n-j)$-dimensional faces, for $j=1,\ldots,(n-1)$.
I will eventually want to talk about densities of sets of such points in the space, and also about measuring the homogeneity or clustering of a distribution of such a set of points.
So you have a set of $3^n$ tuples $(x_1, x_2, \dots, x_n)$ with $x_i \in \{-1,0,1\}$. You can regard these as representing a set $S_n$ of $3^n$ points in $\mathbb R^n$ or as representing the vector space $(\mathbb{Z}/3\mathbb{Z})^n$ of dimension $n$ over the field $\mathbb{Z}/3\mathbb{Z}$.
Note that these are not the same thing. For example, in $\mathbb R^2$ the points in $S_2$ that lie on the line $\sum x_i=1$ are the two points $(1,0)$ and $(0,1)$. But in $(\mathbb{Z}/3\mathbb{Z})^2$ the line $\sum x_i=1$ also includes a third point $(-1,-1)$.