I ask your expertise for the following point:
let P be a polyhedron unbounded and without lines. According to a standard decomposition, P can be written as:
$$P= M + K$$
(+ stands for the Minkowsky sum)
where $M$ is the polytope spanned by its vertices and $K$ is the recession cone of $P$.
My questions is:
Let F be proper face of P
$$F = P \cap H,$$ defined by the plane $$H = \{x\in V : \ell (x) = \alpha\}.$$ If $p$ is a point of $F$, can I say that
$$p = m + u,$$
where
$$m \in M\cap F, \quad u \in H$$
If the sentence is correct, what is the proof the claim?
Thank you very much in advance for your reply