Convex Hulls of Point-Sets and Non-uniform Hypergraphs

Abstract

For fixed integers k ≥ 3 and hypergraphs ${\mathcal G}$ 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 ·(ln T)^{(k − i)/(k − 1)}), i = 3, ..., k, for some number T ≥ 1, we show that the independence number $\alpha ({\mathcal G})$ satisfies $\alpha ({\mathcal G}) = \Omega ((N/T) \cdot (\ln T)^{1/(k-1)})$. Moreover, an independent set I of size |I| = Ω((N/T) ·(ln T)^{1/(k − 1)}) can be found deterministically in polynomial time. This extends a result of Ajtai, Komlós, Pintz, Spencer and Szemerédi 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.