Let $S \subset \mathbb{R}^3$ be a surface, and $p_1$, $p_2$, $p_3$, $p_4$ $\in$ $S$.
If $p_1$, $p_2$, $p_3$ and $p_4$ are close enough, we can define $\gamma(p_i,p_j)$ as the trace of the unique geodesic passing through $p_i$ and $p_j$.
Now define (if possible) $R_{p_1,p_2,p_3,p_4} \subset S$ as the region (an open and connected set) whose boundary satisfies
$$\partial R_{p_1,p_2,p_3,p_4} = \gamma(p_1,p_2) \cup \gamma(p_2,p_3) \cup \gamma(p_3,p_4) \cup \gamma(p_4,p_1).$$
An image to help understand the definition above
I would like to know if given $p$ $\in$ $S$ and $\varepsilon>0$, there always exist $p_1, p_2, p_3, p_4$ $\in$ $\Sigma_p (\varepsilon)$, such that $p \in R_{p_1,p_2,p_3,p_4}$ ?
Where $\Sigma_p (\varepsilon)$ is the geodesic ball centered in $p$ with radius $\varepsilon$.