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 $$\Large F(x,y) , \alpha, \beta, \gamma, \delta) := \frac{\|(\frac{x_1+ y_1}{2}, \frac{x_2+y_2}{2}, x_3, x_4)\| +\|(\frac{x_1+ y_1}{2}, \frac{x_2+y_2}{2}, y_3, y_4)\| }{2 \max (\|(x_1, x_2, x_3, x_4)\|, \|(y_1, y_2, y_3, y_4)\|)}$$