Let $\mathcal L$ be the space of unimodular 2-lattices in $\mathbb R^2$ (namely with covolume $1$). I believe there is a fact about primitive vectors (those who are not nontrivial integer multiple of other vectors) in a lattice saying that (because of the covolume $1$ restriction)
There exists $A>0$ such that a symmetric convex body of area $A$ centered at $O$ contains at most one pair of primitive vectors ($v$ and $-v$) from any unimodular lattice in $\mathbb R^2$.
My question is: what is the smallest/infimum of $A$? Can we explicitly compute it?
I believe I can do this for special convex bodies like balls and rectangles but the general question seems to be hard