Assume $X=\mathbb{C}\mathbb{P}^{n}$, $\omega$ is the Fubini-Study Kähler form and let $B_{R}$ denote the euclidean ball centered at the origin and of radius $R$ in $\mathbb{C}^{n}\subset\mathbb{C}\mathbb{P}^{n}$. Then for $x\in\mathbb{C}^{n}$,

Indeed set $\psi_{R}:=\max(\frac{1}{2}\log[1+||x||^{2}],\varphi_{R})$, where $\varphi_{R}=\frac{1}{2}\log[1+R^{2}]+\log\frac{||x||}{R}$. Recall that the usual Siciak’s extremal function of $B_{R}$ is $\log^{+}\frac{||x||}{R}$. Therefore $\frac{1}{2}\log[1+||x||^{2}]\leq\varphi_{R}=\psi_{R}$ for $||x||\geq R$. On the other hand if $||x||<R$ then $1+||x||^{2}>(1+R^{2})\frac{||x||^{2}}{R^{2}}$ hence $\frac{1}{2}\log[1+||x||^{2}]>\varphi_{R}$ in $B_{R}$.

Now let $u\in PSH(\mathbb{C}\mathbb{P}^{n},\omega)$ such that $u\leq 0$ in $B_{R}$. Then $v=u+\frac{1}{2}\log[1+||x||^{2}]\in{\mathcal{L}}(\mathbb{C}^{n})$. Since $v\leq\frac{1}{2}\log[1+R^{2}]$ in $B_{R}$ we infer $v\leq\frac{1}{2}\log[1+R^{2}]+\log^{+}\frac{||x||}{R}=\psi_{R}$ in $\mathbb{C}^{n}\setminus B_{R}$. Moreover $v\leq\frac{1}{2}\log[1+||x||^{2}]=\psi_{R}$ in $B_{R}$ hence $v\leq\psi_{R}$ in $\mathbb{C}^{n}$. This shows $V_{B_{R},\omega}=\psi_{R}-\frac{1}{2}\log[1+||x||^{2}]$ on $\mathbb{C}\mathbb{P}^{n}$.