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:
* an optimal lower bound for shallow packings,
* improved bounds on Mnets, providing a combinatorial analogue to Macbeath regions in convex geometry,
* we observe that Mnets provide a general, more powerful framework from which the state-of-the-art unweighted epsilon-net results follow immediately, and
* simplifying and generalizing one of the main technical tools in [Fox et al. , J. of the EMS, to appear].
Epsilon-nets
Haussler's packing lemma
Mnets
shallow-cell complexity
shallow packing lemma
38:1-38:15
Regular Paper
Kunal
Dutta
Kunal Dutta
Arijit
Ghosh
Arijit Ghosh
Bruno
Jartoux
Bruno Jartoux
Nabil H.
Mustafa
Nabil H. Mustafa
10.4230/LIPIcs.SoCG.2017.38
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