If source $X$ and target $Y$ are finite spaces of cardinality $m,n$, respectively, then couplings between the uniform measures $\frac{1}{m}1_X$ and $\frac{1}{n}1_Y$ are represented by doubly stochastic $m\times n$ matrices $(a_{ij})$, where every column sums to $1/m$ and every row sums to $1/n$. When $m=n$, it is very important exercise to prove that the set of doubly stochastic matrices is compact convex and has $m!$ extreme points corresponding precisely to permutation matrices. The proof motivates the idea of cyclical monotonicity, which is key definition in OT.

But X+Y does not appear uniform if X, Y uniform. E.g. the sum of two fair dice is not uniform ( 7,8 more frequent than 2, 12).

Levi-Civita's approach in Ch. XI of "The Abs. Dif. Cal" is the clearest GR exposition i've yet found, where he explains the vanishing tensor divergence of $T={T_{ij}}$ as equivalent to a "mass-energy" continuity equation in, with the important caveat that the tensor divergence is defined iff $T$ is tensorial. And the tensoriality of $T$ appears to depend on the molecular media model AND the assumption that there are no forces acting at a distance (Levi Civita posits $F=0$ in vector equation $\rho x'' = \rho F - \chi$, where $\chi=-div\Theta$ is divergence of a stress tensor $\Theta$.)

$GL(\mathbb{Z}^1)\approx \mathbb{Z}^\times=\{\pm 1\}$, where $\mathbb{Z}^\times$ is multiplicative group of units. If $V$ is affine and $G$-invariant, then $G|_V$ is an affine representation of $G$ (more complicated). If $V$ is linear subspace, then your question appears to be: "are all lattice vectors at some uniform distance from a primitive lattice vector?" I think the answer is No, but I might be wrong. Also this question might be answered in a lemma from Borel/Harish-Chandra "Arithmetic Subgroups of Linear Algebraic Groups". Need verify.

If you replace $GL(\mathbb{Z}^n)$ with $GL(\mathbb{Z}^n)$. then there exists a bounded $P$ whose $GL(\mathbb{Q}^n)$ translates cover $V$. [Proof required]

Notice the result is false in one-dimension $n=1$, since there does not exist a bounded subset $P$ of $\mathbb{R}$ for which the $GL(\mathbb{Z}^1)\approx \{\pm 1\}$ translates cover $\mathbb{R}$. Makes me think the result is even false for $GL(\mathbb{Z}^2)$ acting on standard lattice $\mathbb{Z}^2$ in $\mathbb{R}^2$ ...

On the other hand, Levi-Civita himself was aware of Einstein's energy being a pseudo-tensor, and attempting to resolve this issue, offered his own gravitational equations. See personalpages.to.infn.it/~zaninett/projects/storia/…

The infimum $\inf_{a\in A} d(x,a)$ obviously exists since $d\geq 0$. If $d$ is proper, then every minimizing sequence will be contained in a compact ball, and there will exist a convergent subsequence. Without $d$ proper, there is another possibility, using fact that convex hulls of finite subsets is always compact in nonpositive curvature (convex hull of finite subset is the continuous image of a compact simplex). If $d$ nonproper and $A$ noncompact, then i think minimum possibly does not exist.