Let $S_1 = \text{conv}(C_1, ..., C_{n+1})$ and $S_2 = \text{conv}(D_1, ..., D_{n+1})$ be two simplices in $\mathbb{R}^{n}$. Consider the following problem $$ \max \|x\|^2 \hspace{0.5cm} s.t \hspace{0.5cm} x \in S_1 \cap S_2$$
Solution
For an element $x \in S_1 \cap S_2$, one can write $x = \sum_{k=1}^{n+1} \alpha_k \cdot C_k = \sum_{k=1}^{n+1} \beta_k \cdot D_k$ with $\sum \alpha_k = \sum \beta_k = 1$ and $\alpha_k, \beta_k \geq 0$ therefore
$$ \|x\|^2 = \left( \sum_{k=1}^{n+1} \alpha_k \cdot C_k \right)^T \cdot \left(\sum_{k=1}^{n+1} \beta_k \cdot D_k \right) = \sum_{i,j} \alpha_i \cdot \beta_j \cdot C_i^T\cdot D_j = \sum_{i,j} x_{i,j} \cdot e_{i,j}$$ where $x_{i,j} = \alpha_i \cdot \beta_j$ and $e_{i,j} = C_i^T\cdot D_{j}$ Let $X = \begin{bmatrix}x_{1,1}\\ x_{2,1}\\ ... \\ x_{{n+1}, {n+1}} \end{bmatrix}$ and $E = \begin{bmatrix} e_{1,1}& e_{2,1} &... &e_{{n+1}, {n+1}}\end{bmatrix}^T$
Now one can write the optimization problem as follows: $$ \max E^T \cdot X \hspace{0.5cm} s.t \hspace{0.5cm} \begin{cases} \sum_{i} x_{i,j} = \zeta_j\\ \sum_{j} x_{i,j} = \psi_i \\ \sum_{i} \zeta_i = 1\\ \sum_{j} \psi_j = 1\\ \sum_{i} \zeta_i \cdot C_i = \sum_j \psi_j \cdot D_j\\ x_{i,j} \geq 0\\ rank(\bar{X}) = 1 \end{cases}$$ in variables $x_{i,j}, \zeta_i, \psi_j$ for $i,j \in \{1, ..., n+1\}$ where $\bar{X} = \begin{bmatrix} x_{i,j}\end{bmatrix}$ is an $(n+1) \times (n+1)$ matrix
Question
This is a linear problem. Does it have the same solution as the previous one?