# Two Proofs for Shallow Packings

@article{Dutta2015TwoPF,
title={Two Proofs for Shallow Packings},
author={Kunal Dutta and Esther Ezra and Arijit Ghosh},
journal={Discrete \& Computational Geometry},
year={2015},
volume={56},
pages={910-939}
}
• Published 1 December 2016
• Mathematics, Computer Science
• Discrete & Computational Geometry
We refine the bound on the packing number, originally shown by Haussler, for shallow geometric set systems. Specifically, let $$\mathcal {V}$$V be a finite set system defined over an n-point set X; we view $$\mathcal {V}$$V as a set of indicator vectors over the n-dimensional unit cube. A $$\delta$$δ-separated set of $$\mathcal {V}$$V is a subcollection $$\mathcal {W}$$W, s.t. the Hamming distance between each pair $$\mathbf{u}, \mathbf{v}\in \mathcal {W}$$u,v∈W is greater than \delta…

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

• Mathematics
SoCG
• 2017
An optimal lower bound for shallow packings is presented, improved bounds on Mnets are presented, providing a combinatorial analogue to Macbeath regions in convex geometry and Mnets provides a general, more powerful framework.

### A Note on the Size-Sensitive Packing Lemma

We show that the size-sensitive packing lemma follows from a simple modification of the standard proof, due to Haussler and simplified by Chazelle, of the packing lemma.

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