Let $$Q=\{x ∶ Ax ≤ b \}≠∅$$
If $Q = P + C$, where $P$ is a polytope and $C$ is a polyhedral cone, prove that $$\{y|Ay ≤ 0\} = \{y|x + y ∈ Q, ∀ x ∈ Q\}$$ The cone $C = \{y|Ay ≤ 0\}$ is called the characteristic cone of Q.
this will be great if give me hint or Idea ,thanks a lot.
First, let us show that $\{y \mid Ay \leq 0\} \subset \{y \mid x + y \in Q, \forall x \in Q\}$. Consider any $y_0$ for which $Ay_0 \leq 0$. Then $$ x \in Q \Rightarrow Ax \leq b \Rightarrow Ax + Ay_0 \leq b + 0 \Rightarrow A(x+y_0) \leq b \Rightarrow x+y_0 \in Q. $$ Next, let us show the reverse inclusion: $\{y \mid x + y \in Q, \forall x \in Q\} \subset \{y \mid Ay \leq 0\}$. Suppose for the sake of a contradiction that there exists $y_0$ such that $x + y_0 \in Q, \forall x \in Q$ and a row $i$ of $A$ such that $a_i^T y_0 > 0$. Let $m = \min \{b_i - a_i^T x \mid x \in Q\}$ with $x_0 \in \arg \min\{ b_i - a_i^T x \mid x \in Q \}$. However, we know that $x_0 + y_0 \in Q$ which leads to the following contradiction: $$ m \leq b_i - a_i^T (x_0 + y_0) = (b_i - a_i^Tx_0) - a_i^Ty_0 < m.$$ Hence, $\{y \mid x + y \in Q, \forall x \in Q\} \subset \{y \mid Ay \leq 0\}$.