See these pages (the one linked to and the pages reachable by clicking on the "next" links) on computing the diameter of the convex hull of a set of points.
My quick summary:
- compute set of points in convex hull (= O(n log n), the only time you get O(n) is if you sort the list first which takes O(n log n) anyway)
- order along boundary (you get this for free if you use a Graham scan for #1)
- use one of the O(n) diameter algorithms to scan for antipodal points with greatest diameter. Shamos algorithm looks good to me as it's one of the rotating calipers algorithms.
