I am trying to implement the algorithm to find the largest triangle in a convex polygon as described in section 6.1 at the end of page 9 from this paper https://arxiv.org/pdf/1705.11035. After reading it a couple of times, I still don't understands how one is supposed to choose the sub-polygons. In the case of interleaving $T_a$ and $T_m$ I can surmise what is required to get $P'$ and $P''$, but if they are not, it really isn't clear to me.
Depending on the configuration, there could be either one or two sets of three compatible intervals; if there are two, they must interleave. As a result, we construct either one or two smaller polygons $P'$ or $P'$ and $P''$ by directly connecting these intervals.
For the case where they interleave... Say I had a list of vertices in CCW order $P$. Given that the first index of $P$ is $a$, and $b$, $c$ are the indices for the two other vertices of $T_a$, while $m$, $m_b$ and $m_c$ are the vertices of $T_m$ then (borrowing Python syntax for lists):
P' = [P[0], *P[m:b + 1], *P[m_b:c + 1], P[m_c:]]
P'' = [*P[0:m + 1], *P[b:m_b + 1], *P[c:m_c + 1]]
That is:
P' = [a, m, m + 1, ..., b, m_b, m_b + 1, ..., c, m_c, m_c + 1, ...]
P'' = ...
