TITLE: Heaviest Convex Polygon
QUESTION [5 upvotes]: Suppose we have an arbitrary function $f : \mathbb{R}^2 \to \mathbb{R}$.  For any subset $s \subseteq \mathbb{R}^2$, we can define $g_f(s)$ as the integral* of $f$ over the region $s$.  Suppose further that we have access to an oracle that will tell us the value of $g_f(s)$ for any $s$.
Now, restrict our attention to subsets of $s$ that are the convex hull of a given subset of points $\bar x_c \subseteq \{x_1, \ldots, x_N \}$ with $x_i \in \mathbb{R}^2$.  Assuming calls to the oracle are O(1), what is the complexity (in terms of $N$) of finding $\bar x_c^* = \arg \max_{\bar x_c} g_f(conv(\bar x_c))$?  Is there a known algorithm or reduction to a known problem?
EDIT: *Previous statement that Scott answered said "average value" here.

REPLY [6 votes]: I'm assuming the N points are fixed ahead of time.  In that case, it seems to me that you can just use the oracle on each triple of points, since any convex polygon with more than three sides will have average at most the maximum of the averages over triangles in any triangulation.  This gives you O(N^3) at worst.