Double directions of a polyhedron (incomplete)

Dependencies:

1. Polyhedral set and polyhedral cone
2. Double direction of a convex set and double recession space

Let $P = \{x \in \mathbb{R}^n: (a_i^Tx \ge b_i, \forall i \in I) \textrm{ and } (a_i^Tx = b_i, \forall i \in E)\}$ be a non-empty polyhedron. Let $D = \{x \in \mathbb{R}^n: (a_i^Tx = 0, \forall i \in I \cup E)\}$. Then the following are equivalent for any $d \in \mathbb{R}^n$:

1. $\forall x \in P$, $\forall \lambda \in \mathbb{R}$, $x + \lambda d \in P$ (i.e., $d$ is a double direction of $P$).
2. $\exists x \in P$, $\forall \lambda \in \mathbb{R}$, $x + \lambda d \in P$.
3. $d \in D$.

It's easy to see that $(1) \implies (2)$. I will prove $(2) \implies (3)$ and $(3) \implies (1)$.

Proof (missing)

Dependency for:

1. Pointing a polyhedron

Info:

• Depth: 7
• Number of transitive dependencies: 11

Transitive dependencies:

1. Group
2. Ring
3. Field
4. Vector Space
5. Semiring
6. Matrix
7. Cone
8. Convex combination and convex hull
9. Convex set
10. Polyhedral set and polyhedral cone
11. Double direction of a convex set and double recession space