Existence of smallest circle containing a polygon

Question

Given a convex polygon $P$, is there a way to find the smallest (meaning, with the smallest possible radius) circle containing $P$? It seems clear that it must exist and that it must go through (at least) one vertex of $P$. In my mind, it makes sense that it has to go through (at least) two vertices of $P$, but I haven't got a proof yet. Of course, if $P$ is cyclic, we already know what happens, but what about in any other case? Also… is it unique?

Answer 1

We can reduce the problem of finding the smallest circle around a polygon to that of finding the smallest circle around the polygon's vertices only.

Computing the smallest enclosing circle for a set of points is a well-studied problem, and does not need computing the convex hull. Welzl's algorithm (described in the link) does this in randomised linear time. In particular, the smallest circle is unique and passes through at least two vertices of the polygon – three in general, four or more in special cases.

Answer 2

Regarding the three or four "special cases": It is obvious that an unlimited number of vertices will lie on the circle if those points are selected to lie on that circle and any additional vertices lie inside the circle.