Let $\beta\equiv (\beta_0, \beta_1)\in \mathcal{B}\subset \mathbb{R}^2$ with $\mathcal{B}$ compact. $\beta$ is a known vector of parameters.
Let $P_0, P_1, F_0, F_1$ be know parameters, each in $[0,1]$.
Consider the following linear programming problems.
\begin{aligned} \underline{p}(\beta)\equiv \min_{a,b} & [P_0-a]F_0+ [b - P_1]F_1 \\ &\text{ s.t. } \\ & 1) \text{ }a\in [0,1]\text{, }b\in [0,1]\\ & 2) \text{ }Q_0: \{0, \beta_0, \beta_0+\beta_1\} \rightarrow [0,1] \text{ with } Q_0(0)=\frac{1}{2}\text{, } Q_0(\beta_0) = P_0 \text{, } Q_0( \beta_0+\beta_1)= a\\ & \text{ is weakly increasing}\\ & 3) \text{ } Q_1: \{0, \beta_0, \beta_0+\beta_1\} \rightarrow [0,1] \text{ with } Q_1(0)=\frac{1}{2}\text{, } Q_1(\beta_0) = b \text{, } Q_1( \beta_0+\beta_1)= P_1\\ &\text{ is weakly increasing} \end{aligned}
and
\begin{aligned} \bar{p}(\beta)\equiv \max_{a,b} & [P_0-a]F_0+ [b - P_1]F_1 \\ &\text{ s.t. }\\ & 1) \text{ } a\in [0,1]\text{, }b\in [0,1]\\ & 2) \text{ } Q_0: \{0, \beta_0, \beta_0+\beta_1\} \rightarrow [0,1] \text{ with } Q_0(0)=\frac{1}{2}\text{, } Q_0(\beta_0) = P_0 \text{, } Q_0( \beta_0+\beta_1)= a\\ & \text{ is weakly increasing}\\ & 3) \text{ } Q_1: \{0, \beta_0, \beta_0+\beta_1\} \rightarrow [0,1] \text{ with } Q_1(0)=\frac{1}{2}\text{, } Q_1(\beta_0) = b \text{, } Q_1( \beta_0+\beta_1)= P_1\\ &\text{ is weakly increasing} \end{aligned}
Question: Is it true that $\forall p \in [\underline{p}(\beta),\bar{p}(\beta)]$, there exists $\tilde{\beta}\in \mathcal{B}$ such that
\begin{aligned} & 1) \text{ } [P_0-a]F_0+ [b - P_1]F_1 = p \text{ has a solution wrto $a,b$}\\ & 2) \text{ } a\in [0,1]\text{, }b\in [0,1]\\ & 3) \text{ } Q_0: \{0, \tilde{\beta}_0, \tilde{\beta}_0+\tilde{\beta}_1\} \rightarrow [0,1] \text{ with } Q_0(0)=\frac{1}{2}\text{, } Q_0(\tilde{\beta}_0) = P_0 \text{, } Q_0( \tilde{\beta}_0+\tilde{\beta}_1)= a\\ & \text{ is weakly increasing}\\ & 4) \text{ } Q_1: \{0, \tilde{\beta}_0, \tilde{\beta}_0+\tilde{\beta}_1\} \rightarrow [0,1] \text{ with } Q_1(0)=\frac{1}{2}\text{, } Q_1(\tilde{\beta}_0) = b \text{, } Q_1( \tilde{\beta}_0+\tilde{\beta}_1)= P_1\\ &\text{ is weakly increasing} \end{aligned} ? Why yes or not? I think the answer should be yes, but your hint would be very appreciated.