Let $\omega$ be the model conical metric on $\mathbb{C}^n$: $$ \omega_\beta:=\sqrt{-1} \beta^2 |z|^{2\beta-2}dz\wedge d\bar{z}+\sum_{k=2}^n \sqrt{-1}dz_k\wedge d\bar{z}_k, \; \beta\in(0, 1). $$ Then for two points $x, y\in \mathbb{C}^n\setminus \{z=0\}$, what is the expression of $d_\beta(x, y)$?
I can handle the one-dimensional case but get stuck for the general case. Any comments is welcome.