Convex hulls of point-sets and non-uniform hypergraphs

For fixed integers k >= 3 and hypergraphs g on N vertices, which contain edges of cardinalities at most k, and are uncrowded, i.e., do not contain cycles of lengths 2, 3, or 4, and with average degree for the i-element edges bounded by O(Ti-1 center dot (In T)((k-i)/(k-1))), i = 3,...,k, for some number T >= 1, we show that the independence number alpha(g) satisfies alpha(g) = Omega((N vertical bar T) center dot (In T)(1/) (1(k-1))). Moreover, an independent set I of size vertical bar I vertical bar = Omega((N vertical bar T) center dot (In T)(1/(k-1))) can be found deterministically in polynomial time. This extends a result of Ajtai, Komlos, Pintz, Spencer and Szemeredi for uncrwoded uniform hypergraphs. We apply this result to a variant of Heilbronn's problem on the minimum area of the convex hull of small sets of points among n points in the unit square [0, 1](2).