JARTOUX , Bruno

The packing lemma of Haussler states that given a set system $(X, R)$ with bounded VC dimension, if every pair of sets in $R$ are 'far apart' (i.e., have large symmetric difference), then $R$ cannot contain too many sets. This has turned out to be the technical foundation for many results in geometric discrepancy using the entropy method (see...

Given a set system (X,R) such that every pair of sets in R have large symmetric difference, the Shallow Packing Lemma gives an upper bound on |R| as a function of the shallow-cell complexity of R. In this paper, we first present a matching lower bound. Then we give our main theorem, an application of the Shallow Packing Lemma: given a...