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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.

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  • $\begingroup$ Isn't the convex hull of your data such a polygon? $\endgroup$
    – nicoguaro
    Commented Jan 9, 2023 at 23:17
  • $\begingroup$ @nicoguaro Not really...it is possible that the convex hull that you mentioned cannot be represented by a union of triangles in $\mathcal{T}$. $\endgroup$
    – ArGenya
    Commented Jan 10, 2023 at 4:20
  • $\begingroup$ But if you have holes in your mesh then the convex set that you are looking for should be formed by triangles outside your initial set. $\endgroup$
    – nicoguaro
    Commented Jan 10, 2023 at 18:32
  • $\begingroup$ @nicoguaro Oh I should add that the solution set should form a polygon without any holes... $\endgroup$
    – ArGenya
    Commented Jan 10, 2023 at 18:38
  • $\begingroup$ I do not understand what you want to achieve. Would you mind showing us an example? $\endgroup$
    – nicoguaro
    Commented Jan 13, 2023 at 21:59

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