Let $C$ be a convex polyhedron, $\Pi$ a polyhedral complex in $C$ and $f\colon C\to\mathbb{R}$ a piecewise affine function. We say that $\Pi$ and $f$ are compatible if $f$ is affine on each polyhedron of $\Pi$. Alternatively, we say that $f$ is a piecewise affine function on $\Pi$. If the function $f$ is concave, it is said to be strictly concave on $\Pi$ if $\Pi=\Pi(f)$. The polyhedral complex $\Pi$ is said to be regular if there exists a concave piecewise affine function $f$ such that $\Pi=\Pi(f)$.