= Solution
Let $q:N_{\mathbb R}\to N(\tau)_{\mathbb R}=N_{\mathbb R}/\operatorname{span}(\tau)$ be the quotient map. The images $q(\sigma)$ for $\tau\preceq\sigma$ satisfy the face and intersection axioms, so $\operatorname{Star}(\tau)$ is a fan.
To prove completeness, take $\bar v\in N(\tau)_{\mathbb R}$, choose a lift $v$, and choose $w$ in the relative interior of $\tau$. Since $\Sigma$ is complete, each $v+mw$ lies in some cone of $\Sigma$. There are finitely many cones, so one cone $\sigma$ contains $v+m_jw$ for an unbounded sequence $m_j$. Closedness gives $w\in\sigma$. Because $w$ lies in the relative interior of the cone $\tau$ and $\sigma\cap\tau$ is a face of $\tau$, this forces $\tau\preceq\sigma$. Finally,
$$
\bar v=q(v+m_jw)\in q(\sigma).
$$
Thus every point of the quotient lies in the support of $\operatorname{Star}(\tau)$, proving that it is a <complete fan>.
Back to article page
