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.

Note: The question of finding the largest semidisk/sector/circular segment contained in a given convex region C too seems unexplored. It is not hard to see that for a given C, the largest semidisk it contains should necessarily have both end points of the diameter on the boundary of C. Based on this property, a basic algorithm (complexity estimate: O(N^4)) has been proposed at https://nandacumar.blogspot.com/2020/07/largest-semidisk-inside-convex-polygon.html