Let $\mathcal{P}_{i,\rho_i}$ be a polytope in $\mathbb{R}^n$ defined as follows $$ \begin{cases} A_{i,1}^T\cdot x + b_{i,1} \leq \rho_i \newline \vdots \\ A_{i,m} \cdot x + b_{i,m} \leq \rho_i \end{cases}$$ for $i \in \{1,..., N>n\}$ and $A_{i,j}\in \mathbb{R}^n$
For a polytope $\mathcal{P} \subseteq\mathbb{R}^n$, define the set $$ \mathcal{C} = \left\{ \begin{bmatrix} \rho_1 \\ \vdots \\ \rho_N\end{bmatrix}\in \mathbb{R}^N | \bigcap_{k=1}^N \mathcal{P}_{k,\rho_k} \subseteq \mathcal{P} \right\} $$
Question
Is the set $\mathcal{C}$ convex?
Answer attempt
Assume $\mathcal{P} =\{x | A_p^T\cdot x + b_p \leq 0, p \in \{1, ..., M\}\}$. Let $\begin{bmatrix}\rho_1^a \\ \vdots\\ \rho_{N}^a \end{bmatrix} , \begin{bmatrix}\rho_1^b \\ \vdots\\ \rho_{N}^b \end{bmatrix} \in \mathcal{C}$. Therefore $$ m_1 = \max A_p^T\cdot x + b_p \leq 0 \hspace{0.3cm} \text{s.t.} \hspace{0.3cm} x \in \{ A_{i,k}^T\cdot x + b_{i,k} \leq \rho_{i}^a \hspace{0.5cm} \forall i,k\} \\ m_2 = \max A_p^T\cdot x + b_p \leq 0 \hspace{0.3cm} \text{s.t.} \hspace{0.3cm} x \in \{ A_{i,k}^T\cdot x + b_{i,k} \leq \rho_{i}^b \hspace{0.5cm} \forall i,k \} $$ Now one should prove that the maximization problem is smaller