Let $N_{\mathbb{R}}\simeq\mathbb{R}^{n}$ be a real vector space of dimension $n$ and $M_{\mathbb{R}}=N_{\mathbb{R}}^{\vee}$ its dual space. The pairing between $x\in M_{\mathbb{R}}$ and $u\in N_{\mathbb{R}}$ will be alternatively denoted by $\langle x,u\rangle$, $x(u)$ or $u(x)$.

A non-empty subset $C$ of $N_{\mathbb{R}}$ is convex if, for each pair of points $u_{1},u_{2}\in C$, the line segment

$${\overline{u_{1}u_{2}}}=\{tu_{1}+(1-t)u_{2}\mid 0\leq t\leq 1\}$$

is contained in $C$. Throughout this text, convex sets are assumed to be non-empty. A non-empty subset $\sigma\subset N_{\mathbb{R}}$ is a cone if $\lambda\sigma=\sigma$ for all $\lambda\in\mathbb{R}_{>0}$.

The affine hull of a convex set $C$, denoted $\operatorname{aff}(C)$, is the minimal affine space which contains it. The dimension of $C$ is defined as the dimension of its affine hull. The relative interior of $C$, denoted $\operatorname{ri}(C)$, is defined as the interior of $C$ relative to its affine hull. The recession cone of $C$, denoted by $\operatorname{rec}(C)$, is the set

$$\operatorname{rec}(C)=\{u\in N_{\mathbb{R}}\mid C+u\subset C\}.$$

It is a cone of $N_{\mathbb{R}}$. The cone of $C$ is defined as

$$\operatorname{c}(C)={\overline{\mathbb{R}_{>0}(C\times\{1\})}}\subset N_{\mathbb{R}}\times\mathbb{R}_{\geq 0}.$$

It is a closed cone. If $C$ is closed, then $\operatorname{rec}(C)=\operatorname{c}(C)\cap(N_{\mathbb{R}}\times\{0\})$.

Any exposed face of a convex set is a face, and the facets of a convex set are always exposed. However, a convex set may have faces which are not exposed. For instance, think about the four points of junction of the straight lines and bends of the boundary of the inner area of a racing track in a stadium.

For instance, the collection of all faces of a convex set defines a convex subdivision of this set. The collection of all exposed faces of a convex set is a convex decomposition, but it is not necessarily a convex subdivision.

In this text, we will be mainly concerned with the polyhedral case.

For a convex polyhedron, there is no difference between faces and exposed faces.

By the Minkowski-Weyl theorem, polyhedra can be explicitly described in two dual ways, either by the H-representation, as an intersection of half-spaces, or by the V-representation, as the Minkowski sum of a cone and a polytope [Roc70, Theorem 19.1]. An H-representation of a polyhedron $\Lambda$ in $N_{\mathbb{R}}$ is a finite set of affine equations $\{(a_{j},\alpha_{j})\}_{1\leq j\leq k}\subset M_{\mathbb{R}}\times\mathbb{R}$ so that

(3.4)

$$\Lambda=\bigcap_{1\leq j\leq k}\{u\in N_{\mathbb{R}}\mid\langle a_{j},u\rangle+\alpha_{j}\geq 0\}.$$

With this representation, the recession cone can be written as

$$\operatorname{rec}(\Lambda)=\bigcap_{1\leq j\leq k}\{u\in N_{\mathbb{R}}\mid\langle a_{j},u\rangle\geq 0\}.$$

A V-representation of a polyhedron $\Lambda^{\prime}$ in $N_{\mathbb{R}}$ consists in a set of vectors $\{b_{j}\}_{1\leq j\leq k}$ in the tangent space $T_{0}N_{\mathbb{R}}(\simeq N_{\mathbb{R}})$ and a non-empty set of points $\{b_{j}\}_{k+1\leq j\leq l}\subset N_{\mathbb{R}}$ such that

(3.5)

$$\Lambda^{\prime}=\operatorname{cone}(b_{1},\dots,b_{k})+\operatorname{conv}(b_{k+1},\dots,b_{l})$$

where

$$\operatorname{cone}(b_{1},\dots,b_{k}):=\bigg\{\sum_{j=1}^{k}\lambda_{j}b_{j}\bigg|\ \lambda_{j}\geq 0\bigg\}$$

is the cone generated by the given vectors (with the convention that $\operatorname{cone}(\emptyset)=\{0\}$) and

$$\operatorname{conv}(b_{k+1},\dots,b_{l}):=\bigg\{\sum_{j=k+1}^{l}\lambda_{j}b_{j}\bigg|\ \lambda_{j}\geq 0,\ \sum_{j=k+1}^{l}\lambda_{j}=1\bigg\}$$

is the convex hull of the given set of points. With this second representation, the recession cone can be obtained as

$$\operatorname{rec}(\Lambda^{\prime})=\operatorname{cone}(b_{1},\dots,b_{k}).$$

There are two natural processes for linearizing a polyhedral complex.

It is natural to ask whether the recession or the cone of a given polyhedral complex is a complex too. The following example shows that this is not always the case.

Therefore, to assure that $\operatorname{rec}(\Pi)$ or $\operatorname{c}(\Pi)$ are complexes, we need to impose some condition on $\Pi$. This question has been addressed in [BS10]. Because our applications, we are mostly interested in the case when $\Pi$ is complete. It turns out that this assumption is enough to avoid the problem raised in Example 3.8.

We consider now an integral structure in $N_{\mathbb{R}}$. Let $N\simeq\mathbb{Z}^{n}$ be a lattice of rank $n$ such that $N_{\mathbb{R}}=N\otimes\mathbb{R}$. Set $M=N^{\vee}=\operatorname{Hom}(N,\mathbb{Z})$ for its dual lattice so $M_{\mathbb{R}}=M\otimes\mathbb{R}$. We also set $N_{\mathbb{Q}}=N\otimes\mathbb{Q}$ and $M_{\mathbb{Q}}=M\otimes\mathbb{Q}$.

Observe that any rational polyhedron admits an H-representation with integral coefficients.