"Let $P$ be a convex $n$-gon and $q$ a point in the plane. Find an algorithm to compute the longest chord whose supporting line contains q."
When $q$ is external to $P$, I think I can prove the longest chord contains at least one vertex of $P$. And for each vertex $v$ the line segment $[qv]$ intersects $P$ at most once (without counting $v$ and eliminating the case where $[qv]$ is colinear with an edge of $P$). An algorithm could be for each vertex to compute its distance with the potential intersection. $\mathcal{O} (n)$ for the loop on vertices, $\mathcal{O} (n\log n)$ to find the intersection with Bentley-Ottman. Overall running time: $\mathcal{O} (n^2\log n)$.
I am wondering if I cannot get a faster algorithm with finding the intersection in $\mathcal{O} (\log n)$, using the convexity to come up with a binary search.
When $q$ is inside of $P$, I am not even sure the chord we are looking for has to contain any vertices of $P$! So I do not even know how to discretize the problem.
The chord doesn't have to contain a vertex of $P$, as this example shows:
$AB$ is clearly longer than $CD$.