Before revision:  Any such matching is called a of P .
After revision:  Any such matching is called a the longest segment. We develop a range of tools for dealing with non-crossing matchings of points in convex position, which enable us to solve the problem of finding bottleneck matchings of points in convex position in O(n^2) time. Combining the same set of tools with a geometric analysis, we design an O(n)-time algorithm for the case where the given points lie on a circle. Previously best known results were O(n^3) for points in convex position, and O(n \log n) for points on a circle .