Hi,
let $L$ be a lattice isomorphic to $\mathbb{Z}^r$ for some positive integer $r$ and $E=L\otimes \mathbb{R}$. An integral paving in $E$ is a set $\Sigma$ of integral polytopes (the vertices are in $L$) $N\subset E$ with the following properties:
1) given $A,B\in\Sigma$ then $A\cap B\in\Sigma$ and $A^0\cap B^0=\emptyset$, where $()^0$ denotes the relative interior
2) if $A\in \Sigma$ and $F\subset A$ is a face then $F\in \Sigma$
3) for any bounded subset $W\subset E$ only finitely many $A\in\Sigma$ have non empty intersection with $W$
a paving $\Sigma$ is called $L$-invariant if it is $L$-translation invariant.
Example: given $q$ a positive quadratic form on $E$ one can look at the convex hull of points ${(l,q(l))}_{l\in L}$. This gives the graph of a picewise linear function $m:E\rightarrow \mathbb{R}$ and the domains of linearity of $m$ give an $L$-invariant integral paving.
Call a paving regular if it comes from a positive definite quadratic form.
My question: assume I have given an integral $L$-invariant paving, is there any criterion to estabilish when this paving is regular?
