TY - GEN
T1 - Shallow packings, semialgebraic set systems, macbeath regions, and polynomial partitioning
AU - Dutta, Kunal
AU - Ghosh, Arijit
AU - Jartoux, Bruno
AU - Mustafa, Nabil H.
N1 - Publisher Copyright:
© Kunal Dutta, Arijit Ghosh, Bruno Jartoux, and Nabil H. Mustafa.
PY - 2017/6/1
Y1 - 2017/6/1
N2 - 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).
AB - 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).
KW - Epsilon-nets
KW - Haussler's packing lemma
KW - Mnets
KW - Shallow packing lemma
KW - Shallow-cell complexity
UR - http://www.scopus.com/inward/record.url?scp=85029947864&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.SoCG.2017.38
DO - 10.4230/LIPIcs.SoCG.2017.38
M3 - Conference contribution
AN - SCOPUS:85029947864
VL - 77
T3 - Leibniz International Proceedings in Informatics, LIPIcs
SP - 38:1-38:15
BT - 33rd International Symposium on Computational Geometry, SoCG 2017
A2 - Katz, Matthew J.
A2 - Aronov, Boris
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 33rd International Symposium on Computational Geometry, SoCG 2017
Y2 - 4 July 2017 through 7 July 2017
ER -