Extremal Connectors for Disjoint Circles

Abstract

We discuss a number of geometric extremal problems for configurations of points on several circles in the plane. Circles are assumed to be nested or have disjoint interior domains. For disjoint circles, we study the problem of minimal connecting cycle and Morse theory of perimeter function. It is shown that minimal connecting cycle is unique if the pairwise convex hulls of circles do not intersect any other circle. For concentric circles, the main attention is given to the critical points of perimeter considered as a function on the product of concentric circles. In this setting, we prove that aligned configurations are nondegenerate critical points of perimeter and give formulas for their Morse indices. If the number of circles does not exceed 4 we prove that the perimeter is a Morse function and describe the shape of maximal connecting cycles. Similar problems are studied for the oriented area of connecting cycle. In conclusion we briefly discuss some possible generalizations.