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:

Transitive dependencies:

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