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I'm studying Frobenius splitting of toric varieties based on Sam Payne's characterization of Frobenius splitting ("Frobenius splittings of toric varieties" (2008)) and I came across a sort of Helly's theorem-type-problem.

Fix an integer $q \geq 2$. Suppose we have a collection $V_{\rho}$ of $n$-dimensional primitive lattice vectors. For each $n$-tuple $\sigma$ of $n$-dimensional vectors in $V_{\rho}$, let $\mathbb{F}_{\sigma} = \bigcap_{v_{\rho} \in \sigma}\{|v_{\rho}\cdot x| < 1\}$. Suppose $\mathbb{F}_{\sigma}$ contains a representative for the residue class $x + (1/q)\mathbb{Z}^{n}/\mathbb{Z}^{n}$ for each $\sigma$. I'm interested in whether the intersection over all $n$-tuples, $\bigcap_{\sigma}\mathbb{F}_{\sigma}$ contains a of representative for $(1/q)\mathbb{Z}^{n}/\mathbb{Z}^{n}$.

Does anyone know of any Helly-type theorems for lattices and specific convex shapes like centrally-symmetric parallelepipeds?

I've looked in J.Hammer's book "Unsolved problems concerning lattice points" from 1977 and the following papers: "A fractional Helly theorem for convex lattice sets" (2003) by $I$. I. Bárány and J. Matousek, "Convexity in Cristallographical Lattices" by Jean-Paul Diognon (1973), and "Helly's theorem (1921) and its relatives" by Ludwig Danzer, Branko Grunbaum, and Victor Klee. These sources don't have quite what I'm looking for and I'm hoping someone can show me where I might be able to find such theorems. Thanks!

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