Convex Hulls of Point-Sets and Non-uniform Hypergraphs

For xed 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(T i 1 (lnT )(k i)=(k 1)), i = 3; : : : ; k, for some number T 1, we show that the independence number (G) satis es (G) = ((N=T ) (lnT )1=(k 1)). Moreover, an independent set I of size jIj = ((N=T ) (lnT )1=(k 1)) can be found deterministically in polynomial time. This extends a result of Ajtai, Koml os, Pintz, Spencer and Szemer edi 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.