Dear all, recently, I encountered the following problem. It is closely related to the order of growth for UMD constants of all $n$-dimensional Banach lattice.
Let $\alpha^k \in (\alpha_1^k, \alpha_2^k, \alpha_3^k, \alpha_4^k) \in [0,1]^4$ for $k = 1, 2, 3, 4$. For $x = (x_1, x_2, x_3, x_4) \in [0,1]^4$, define $$\| x\|_k = \sum_i \alpha_i^k x_i.$$
Then define $$\| x\| = \max(\| x \|_1, \| x \|_2, \|x \|_3, \| x \|_4).$$
Define $$\Large F(x,y, \alpha, \beta, \gamma, \delta) := \frac{\|(\frac{x_1+ y_1}{2}, \frac{x_2+y_2}{2}, x_3, x_4)\| +\|(\frac{x_1+ y_1}{2}, \frac{x_2+y_2}{2}, y_3, y_4)\| }{2 \max (\|(x_1, x_2, x_3, x_4)\|, \|(y_1, y_2, y_3, y_4)\|)}$$
I want to solve the following: find $(x_0, y_0, \alpha_0, \beta_0, \gamma_0, \delta_0)$, $$F(x_0, y_0,\alpha_0, \beta_0, \gamma_0, \delta_0) = \Large \max F(x,y, \alpha, \beta, \gamma, \delta) $$ $$\Large \textbf{s.t.} \alpha, \beta, \gamma, \delta\in [0,1]^4, \quad x, y \in [0, 1]^4, x \ne 0.$$
This is equivalent to $$\max\frac{\|(\frac{x_1+ y_1}{2}, \frac{x_2+y_2}{2}, x_3, x_4)\| +\|(\frac{x_1+ y_1}{2}, \frac{x_2+y_2}{2}, y_3, y_4)\| }{2} $$ $$\Large \textbf{s.t.} \alpha, \beta, \gamma, \delta\in [0,1]^4, \quad x, y \in [0, 1]^4, x \ne 0$$ $$ \Large\|(x_1, x_2, x_3, x_4)\| \le 1$$ $$\Large\|(y_1, y_2, y_3, y_4)\| \le 1$$
Probably, this problem is classical in convex program, and maybe we even have software for computing this kind of problem. Are there anyone who can do this by computer (or maybe even by hand)?
$\textbf{Remark}:$ I have tried some example by hand, and I know that $ \max F(x,y) \ge \frac{3}{2}$. Theoretically, I also know that $\max F(x,y) \le 2$. The most interesting question is: Do we have $$ \max F(x,y) = 2 ?$$ If it is, I can get an optimal estimate of the supremum of (say) $\text{UMD}_2$ constant for all $n$-dimensional Banach lattice.
