Geometrical and analytical characterizations of piecewise affine mappings

Let $X$ and $Y$ be finite dimensional normed spaces, $\mathcal{F}(X,Y)$ a collection of all mappings from $X$ into $Y$. A mapping $P\in\mathcal{F}(X,Y)$ is said to be piecewise affine if there exists a finite family of convex polyhedral subsets covering $X$ and such that the restriction of $P$ on each subset of this family is an affine mapping. In the paper we prove a number of characterizations of piecewise affine mappings. In particular we prove that a mapping $P\colon X\to Y$ is piecewise affine if and only if for any partial order $\preceq$ defined on $Y$ by a polyhedral convex cone both the $\preceq$-epigraph and the $\preceq$-hypograph of $P$ can be represented as a union of finitely many of convex polyhedral subsets of $X\times Y$. Without any restriction of generality we can suppose in addition that the space $Y$ is ordered by a minihedral cone or equivalently that $Y$ is a vector lattice. Then as is well known the collection $\mathcal{F}(X,Y)$ endowed with standard pointwise algebraic operations and the pointwise ordering are a vector lattice too. In the paper we show that the collection of piecewise affine mappings coincides with the smallest vector sublattice of $\mathcal{F}(X,Y)$ containing all affine mappings. Moreover we prove that each convex (with respect to an ordering of $Y$ by a minihedral cone) piecewise affine mapping is the least upper bound of finitely many of affine mappings. The collection of all convex piecewise affine mappings is a convex cone in $\mathcal{F}(X,Y)$ the linear envelope of which coincides with the vector subspace of all piecewise affine mappings.