Shallow Packings, Semialgebraic Set Systems, Macbeath Regions, and Polynomial Partitioning

The packing lemma of Haussler states that given a set system (X,R) with bounded VC dimension, if every pair of sets in R have large symmetric difference, then R cannot contain too many sets. Recently it was generalized to the shallow packing lemma, applying to set systems as a function of their shallow-cell complexity. In this paper we present several new results and applications related to packings: 1. an optimal lower bound for shallow packings, 2. improved bounds on Mnets, providing a combinatorial analogue to Macbeath regions in convex geometry, 3. we observe that Mnets provide a general, more powerful framework from which the state-ofthe-art unweighted -net results follow immediately, and 4. simplifying and generalizing one of the main technical tools in Fox et al. (J. of the EMS, to appear). 1998 ACM Subject Classification F.2.2 Nonnumerical algorithms and problems, G.2.1 Combinatorics