I have a set of points $C_i$ on a two dimensional plane and I want to find a point $P$ such that the maximum distance from $P$ to any of the points is minimised, i.e. minimise(max($||P-C_i||$)).
I've found that the centroid of $C_i$ gives a reasonable approximation, but does not find the optimal solution for $P$. I've also tried a few other options like finding the centroid of the convex hull of $C_i$ and they have all produced comparable or worse results.
I've come to the realisation that there's almost certainly not an analytic solution to my case, so I'm looking for an algorithm that would be faster than just doing a grid search around the area near the centroid.