Given a set of N points in general position on the plane, the problem is to give efficient algorithms to find

1. the smallest semicircular region (semidisk) that contains the points
2. the smallest circular segment that contains the points (2 variants - 'smallest' could mean either of 'least area' or 'least perimeter').
3. the smallest sector that contains the points (again, 2 variants)

I am not aware of previous works on these questions.

Note: An O(N^2) algorithm for question 1 has been proposed 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.

Higher dimensional analogs of these questions - hemisphere, spherical segment etc..- arise naturally.