Special coordinates for periodic metrics

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?

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In general, Voronoi cells need not be contractible (locally, there are no restrictions on the metric $g$, which I assume to be Riemannian). If you assume that $g$ is flat then the answer is trivially positive, of course. –  Misha Mar 21 '13 at 15:23
@Misha: You have a fair point. What's left of my question is therefore pretty vague, I'll try to precise it a little bit. –  Benoît Kloeckner Mar 21 '13 at 17:24
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1 Answer

Maybe this can help, but it might not be what you want: Add reflections at affine planes $\{x: x_i = \mathbb Z +\frac12\}$, then the $\mathbb Z^n$ action is extended to an action which is generated by reflections. Make your $\mathbb Z^n$-invariant metric also invariant under the extended reflection group (you have to average over the $n$ generating reflections only) and try to use methods from the following two papers (on the arXiv or on my homepage):

• Dmitri Alekseevky, Andreas Kriegl, Mark Losik, Peter W. Michor: Reflection groups on Riemannian manifolds. Annali di Matematica 186 (2006), 25-58

• Dmitri V. Alekseevsky, Peter W. Michor, Yurii A. Neretin: Rolling of Coxeter polyhedra along mirrors. 19 pages. To appear in: Geometric Methods in Physics: XXXI Workshop, Bialowieza, Poland, June 24–30, 2012. Trends in Mathematics, Birkhauser, Due: June 30, 2013. arXiv:0907.3502.

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