Consider a convex polygon in $\mathbb{R}^2$ with multiple convex holes in it and suppose that, for now, we have a 2D triangular mesh of the polygon, which is represented by $\mathcal{T} \equiv\{T_i\}_i$, a set of triangles in $\mathbb{R}^2$. My question is, given a triangle $T$ that lies at the exterior boundary of the polygon, is there a way to find a set $\{T_{i_j}\}_j \subset \mathcal{T}$ such that
$$ \left(\bigcup_j T_{i_j}\right)\cup T $$
forms the largest convex polygon (without any holes) that contains $T$? Any help or hint will be greatly appreciated!
My Attempts:
I first wanted to solve the problem with a greedy algorithm. That is, check whether the union between $T$ and (not necessarily all) other triangles that share edges with $T$ is convex. If so, combine them and repeat this step with the resulting shape, and terminate when no more such triangles can be merged. But I find it hard to guarantee that the final resulting shape will be maximal.
