Locating the Center of a Set of Points on a Curve

Given is a finite set of points, A, on a simple closed curve. Henceforth, point refers to an arbitrary point on the curve, and a point in A is called an anchor. For any two points x, y, the distance between them, d(x, y), is the length of the shorter segment (of the curve) joining x, y. The metric of a point with respect to the given anchors is the sum of the distances between the anchors and the point, i.e., for a point p, its metric with respect to A, M(p,A), is ∑ y∈A d(p, y). It is required to find a point with the smallest metric; we call such a point a center of the given set of anchors, and we denote its metric by M(A). First, we solve the problem when the anchors are on a simple open curve; in that case, there is a unique segment of the curve joining any two points. We give a simple characterization of the center in this case. Next, we use the result for open curves to locate the center in a closed curve.