This question is a follow-up to that one.

Given a $\mathbb{Z}^n$-periodic metric $g$ on $\mathbb{R}^n$ (with $n>2$), is it possible to find a periodic diffeomorphism $\varphi$ such that $\varphi^*g$ makes the voronoi cell of $\mathbb{Z}^n$ convex? Or more generally, what kind of good compatibility between the metric and the affine structure of $\mathbb{R}^n$ can one expect by choosing good coordinates on the quotient torus?

**Edit.** Misha's comment shows that the precise part of the question is very naive. To make the remaining part more precise, one "compatibility" that would do for my need would be the following.

Call $g$ "$k$-balanced" if for all $v$ in the Voronoi cell of $0$, we have $$\sup_{p\in\mathbb{R}^n} d(p,p+v) \le k \ \mathrm{diam}(g)$$ Is it true that there is a $k=k(n)$ such that for all periodic riemmannian metric $g$, there is a periodic diffeomorphism $\varphi$ such that $\varphi^*g$ is $k$-balanced?