Let $C$ be a full-dimensional rational polyhedral cone in $\Bbb R^d$ with facets $G_1,\ldots,G_n$ . For each $i$, let $h_i$ be an integer-valued linear functional on $\Bbb R^d$ whose kernel is the span of $G_i$. Here's my question: Choose a subset of the facets. Is it possible to find a point $x$ in $\Bbb R^d$ such that $h_i(x)\in \Bbb Z$ iff $G_i$ is one of our chosen facets?
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Clearly yes if $d\leq 2$. Clearly yes if $n\leq 3$, since the linear functionals $h_i$ are independent. Clearly no if $d>2$ and $n\geq 4$, as an example $x\geq 0$, $x+y\geq0$, $x+z\geq 0$, $y+z\geq 0$ shows (if $x$, $x+y$, and $x+z$ are integral then $y+z$ is integral as well).
