Given an expression in the Max-Plus algebra is it possible to recognize if it represents a continuous piecewise linear (CPWL) function whose polyhedral complex is a hyperplane arrangement?
Or conversely is every CPWL function whose polyhedral complex is a hyperplane arrangement guaranteed to have a certain specific representation in the Max-Plus algebra?
There is one answer known to the general version of the second question in the following form (as in, the following doesn't seem to see anything about hyperplane arrangements!):
That for every continuous piecewise linear function $f :\mathbb{R}^n \to \mathbb{R}$, there exists a finite set of affine linear functions $\ell_1, \ldots, \ell_k$ and subsets $S_1, \ldots, S_p \subseteq \{1, \ldots, k\}$ (not necessarily disjoint) where each $S_i$ is of cardinality at most $n+1$, such that \begin{equation}\label{eq:hinged-hyperplane}f = \sum_{j = 1}^p s_j \bigg(\max_{i \in S_j} \ell_i\bigg),\end{equation} where $s_j \in \{-1,+1\}$ for all $j=1, \ldots, p$.