Handouts:
3/12/08: Convex hulls in 2D: last remarks. Convex polytopes and convex hulls in 3 or more dimensions

Review Melkman's convex hull algorithm: It computes in linear time (O(n)) the convex hull of a polygonal chain that is SIMPLE (not self-intersecting). (Equivalently, it computes the convex hull of a set of points that are given in an order such that linking the points in that order yields a noncrossing polygonal chain.) The main point is that SIMPLICITY HELPS: instead of worst-case O(n log n) or O(n log h), we can get O(n) if we know the points are the vertices of a simple chain.
I go over detailed examples of applying Melkman's algorithm, and a sketch of the proof of correctness and the running time.
I remark about a relatively new result (my own work, joint with Levcopoulos and Lingas) that shows that a convex hull of n points can be found in time O(n log k), where k is the number of subchains in a partition of the input chain into the fewest simple subchains. (Thus, if the input points specify a simple chain, k=1, and it takes time O(n); if the input chain arbitrarily intersects intself, k is about n, and the algorithm takes time O(n log n).)
I started to introduce convex hulls in 3 or more dimensions, stating that the time it takes to compute a hull in d dimensions for d>=4 is O(n^{floor(d/2)}), which is the worst-case complexity of the output size (ie., the hull may have that many faces).
Combinatorics: planar graphs, Euler's formula, linearity of the number of edges and faces (as a function of n, the number of vertices).