# Zoll Flat Finsler tori and convex bodies on a starry night

The starry night. The "celestial sphere" is given by set of non-zero vectors in $\mathbb{R}^n$ modulo positive dilations (i.e., $v \equiv w$ if $v = \lambda w$ for some $\lambda > 0$) and the "stars" are given by the non-zero elements of the integer lattice $\mathbb{Z}^n$ modulo the same equivalence relation.

Problem. Given a hypersurface $M \subset \mathbb{R}^n$ bounding a convex body, consider its set-valued Gauss map as taking values on (the power set of) the celestial sphere. Assume that the images of points where the Gauss map is single-valued are "stars", does this imply that $M$ is the boundary of a polytope?

Remark that by Rademacher's theorem the set of points where the Gauss map is single-valued has full measure on $M$. Likewise, by Alexandrov's theorem almost every point of $M$ has an osculating quadric: in this case it will be a degenerate quadric. However, I don't think this means that we have a polytope (aren't Yves Benoist's convexes divisibles all like this in that they have curvature zero everywhere, but are strictly convex ??).

Motivation. This problem is related to the construction of flat tori almost all of whose geodesics are closed and of the same length. Somewhat surprisingly such tori exist: consider the Hamiltonian $H : T^* T^2 \rightarrow [0,\infty)$ given by $$H(q_1,q_2;p_1,p_2) = |p_1| + |p_2|$$ and the associated $L_\infty$ Finsler metric on the torus. If one considers the equations of motion on the (open dense) set of the unit cosphere bundle where the Hamiltonian is smooth, we see that for every initial condition we obtain only closed geodesics of the same length.

To obtain other Zoll flat tori, we just need to find norms $H$ (the Hamiltonian of the metric will be $(q,p) \mapsto H(p)$) for which $dH(p)$ is a primitive element of the integer lattice at every point of the hypersurface $H = 1$ where $H$ is differentiable. Dually, we must take polytopes that contain the origin as an interior point and for which all vertices are primitive elements of $\mathbb{Z}^n$.

If we just require that almost all geodesics be closed and drop the condition on the periods, we arrive at the problem stated above. Sergei's answer shows that in this case we can have norms whose unit balls are not polytopes.

Zoll tori come up naturally in the systolic study of Finsler tori and the geometry of numbers which is the subject of a forthcoming paper by Balacheff,Tzanev, and myself.

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No such a set is not always a polytope. Consider the convex hull of the set of points of the form $(1/n.1/n^2)$ and $(0,0)$ in $\mathbb R^2$. Its boundary is a union of infinitely many segments with rational directions. And all points not in the interiors of the segments are "vertices" where the Gauss map is not single-valued.