Consider $(\mathbb{C}P^2,\omega\_{FS})$ where $\omega\_{FS}$ is the standard Fubini-Study form. Let $L$ denote a sphere in $\mathbb{C}P^2$ in the class $\mathbb{C}P^1$. Further let $\int\_{L} \omega\_{FS} = \pi$. 


Then the map 


$$\begin{align} 
i : (B(1),\omega\_0) &\to (\mathbb{C}P^2 \setminus L, \omega\_{FS}) \\
 (z,w) &\mapsto [\sqrt{1 - |z|^2 -|w|^2} : z : w]
\end{align}$$


is a symplectomorphism. Where $(B(1),\omega\_0)$ denotes the standard ball of radius 1 in $\mathbb{R}^4$ with the restriction of the symplectic standard form in $\mathbb{R}^4$. 


Let $B\_{\mathbb{C}P^2}(x\_0, r)$ denote a metric ball in $\mathbb{C}P^2$ (for standard Kahler metric) of radius r and centred at $x\_0 \in L$.


Then the paper I'm reading claims that $i^{-1} ( \mathbb{C}P^2 \setminus (L \cup B\_{\mathbb{C}P^2}(x\_0, r))$ is a star shaped neighbourhood in $B(1)$. 


Could anyone help me prove the above statement?



Star-shaped domain in $\mathbb{C}P^2$
One way to prove this is as follows.


First, from the assumption $B(1)\subset \mathbb C^2$ and the centre of $B(1)$ is $(0,0)$. Now we need the following two claims.


*Claim 1.* Any straight geodesic unit segment $I$ in $B(1)$ going through $(0,0)$ is sent by $i$ to a geodesic segment in $\mathbb CP^2$ through the point $i(0,0)$ in $\mathbb CP^2$. 


*Claim 2.* Let us parametrise $I$ by its length for $t\in [0,1]$ let $p(t)$ be the point on $I$ on distance $t$ from $(0,0)\in I$. The the distance function $d\_{\mathbb CP^2}(x\_0, i(p(t)))$ is monotone decreasing for $t\in [0,1]$.


Clearly, it follows from Claims 1, 2, that the domain you consider is star-shaped with respect to $(0,0)$. If you want to prove Claim 2, one way to do this is to represent $\mathbb CP^2$ as the metric quotient $S^5/S^1$ and understands what Claim 1 means for this representation. Here $S^5$ is the unit sphere in $\mathbb C^3$ and $S^1$ is acting by multiplication by complex units. Claim 1 is rather simple. One can check it for the segment tangent to the vector $(1,0)\subset T\_{0,0}(\mathbb C^2)$ and then use the fact that the map $i$ is $U(2)$ equivariant. 


