Shallow packings, semialgebraic set systems, macbeath regions, and polynomial partitioning

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Conference Article

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Leibniz International Proceedings in Informatics, LIPIcs


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-of-the-art unweighted e-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).

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