Talk:Convex hull

The following paragraph was in the convex article, but since it's about convex hulls it would be better suited to this convex hull article. I'm leaving it on the Talk page for now, however, as I don't think it would add anything useful to this article either. --Zundark 10:43, 16 Dec 2003 (UTC)

One application of convex hulls is found in efficiency frontier analysis. Efficiency is assumed to be a monotonic function of each of finitely many real variables. Each one of finitely many data points is in exactly one hull, and is considered more efficient than all data points in hulls contained within its own hull. A particle whose velocity vector has a value of a for all coordinates representing maximized variables, and a value less than a for all minimized variables, will pass through the hulls in increasing order of efficiency.

The Convex Hull article is inaccurate. Convex Hulls of 2D polygons are clearly Omega(n log n) by reduction to sorting: pick values x_i, compute convex hull of (x_i, x_i^2). Find point with smallest x, output the rest in order. Since a parabola is convex, the convex hull of the subset of the points will have all of them. For that to be the case, the output permutation will be such that points are output in order. If convex hulls were possible in Omega(n), then it would be possible to sort points in Omega(n). By a decision tree model, sorting points is Omega(n log n), and so is the convex hull.

You better read more carefully. First, The O(n) is not for 2D polygons, but for 2D simple polygons. Second, how do you make a polygon from points (x_i, x_i^2)? mikka (t) 02:10, 20 July 2005 (UTC)

Connect (x_1, x_1^2) to (x_n, x_n^2), and you'll have a simple convex polygon. And for every non-simple polygon, there is a simple polygon with the same vertex set, so the simple/non-simple polygon distinction is pointless (because convex hull really only cares about the vertex set)

You are wrong. Sorry, I have no intentions to teach you math. get yourself a book in computational geometry. mikka (t)

As much as I love to reply to ad hominems, could you please show me the flaw in my argument? And, by the way, the reduction proof is in Cormen, Leiserson and Rivest.