Given a set of N points in general position on the plane, the problem is to give efficient algorithms to find
- the smallest semicircular region (semidisk) that contains the points
- the smallest circular segment that contains the points (2 variants - 'smallest' could mean either of 'least area' or 'least perimeter').
- the smallest sector that contains the points (again, 2 variants)
I am not aware of previous works on these questions. I am seeking relevant pointers to the literature.
Full disclosure: I proposed an O(N^2) algorithm for question 1 in https://arxiv.org/abs/2005.10245. No proof of optimality was given, so there could be faster algorithms.
Some thoughts on question 2 are also given there but no algorithm.