Generalized Circumcenter: minimizing the range of distances from a point to the vertices of a polygon

Question

It is well known that the circumcenter of a polygon exists if and only if the polygon is cyclic.

I would like to extend the definition of a circumcenter for noncyclic polygons.

Let us define $c(A)$ as the range of the lengths of the distances from $A$ to the vertices of the polygon; that is, the longest minus shortest distance from $A$ to the vertices of the polygon.

 The range is chosen as a simple measure of spread.

If there exists an $A_0$ such that $0 \leq c(A_0) < c(A)$ for all $A$ not equal to $A_0$, and $A_0$ is not equivalently at infinity, then this $A_0$ is defined to be the generalized circumcenter of the polygon.

Note that this generalization follows from the fact that the distances from the circumcenter to the vertices of a cyclic polygon are equal to each other.

Does the generalized circumcenter exist for all n-gons?

Answer 1

Assuming by "range" you mean the maximum distance from $A_0$ to the vertices of the polygon, your "generalized circumcenter" is actually the center of the minimum covering circle

It is indeed unique: if there were two such points and circles with the same radius all the points then they would overlap and the a circle centered on that overlap would have a strictly smaller radius

It is not really a circumcenter. For example with an obtuse-angled triangle, the minimum covering circle is smaller than the circumcircle, the center of the minimum covering circle is the midpoint of the longest edge, and the minimum covering circle only passes through two of the vertices

Answer 2

No, there are some polygons for which the GCC (generalized circumcenter), as you have defined it, does not exist.

Counterexample: consider an irregular pentagon whose 5 vertices consist of one point at the origin, followed by the vertices of a square centered at the origin: $$ p_0 = (0,0) \\ p_1 = (1,0) \\ p_2 = (0,1) \\ p_3 = (-1,0) \\ p_4 = (0,-1) \\ $$ Since your definition of GCC depends only on the vertex set without caring about ordering, the GCC, if it exists, must be the symmetry center of the point set; i.e. it must be the origin, whose range to the vertices is $1-0=1$. But there are other points who also have range $1$. E.g. $A = (0.1, 0)$ has range $c(A) = 1.1 - 0.1 = 1$.