Shallow Packings in Geometry

TL;DR

This paper refines bounds on the packing number for shallow geometric set systems, providing tighter estimates that improve understanding of geometric configurations and their combinatorial complexity.

Contribution

It introduces a generalized bound on the packing number for shallow set systems with bounded primal shatter dimension, extending Haussler's classical results.

Findings

Refined the bound on the packing number for shallow geometric set systems.

Showed that set systems of halfspaces, balls, and slabs have better packing bounds when the parameter k is smaller.

Applied the new bounds to problems in spanning trees and geometric discrepancy.

Abstract

We refine the bound on the packing number, originally shown by Haussler, for shallow geometric set systems. Specifically, let $\V$ be a finite set system defined over an $n$-point set $X$; we view $\V$ as a set of indicator vectors over the $n$-dimensional unit cube. A $\delta$-separated set of $\V$ is a subcollection $\W$, s.t. the Hamming distance between each pair $\uu, \vv \in \W$ is greater than $\delta$, where $\delta > 0$ is an integer parameter. The $\delta$-packing number is then defined as the cardinality of the largest $\delta$-separated subcollection of $\V$. Haussler showed an asymptotically tight bound of $\Theta((n/\delta)^d)$ on the $\delta$-packing number if $\V$ has VC-dimension (or \emph{primal shatter dimension}) $d$. We refine this bound for the scenario where, for any subset, $X' \subseteq X$ of size $m \le n$ and for any parameter $1 \le k \le m$, the number of vectors of length at most $k$ in the restriction of $\V$ to $X'$ is only $O(m^{d_1} k^{d-d_1})$, for a fixed integer $d > 0$ and a real parameter $1 \le d_1 \le d$ (this generalizes the standard notion of \emph{bounded primal shatter dimension} when $d_1 = d$). In this case when $\V$ is "$k$-shallow" (all vector lengths are at most $k$), we show that its $\delta$-packing number is $O(n^{d_1} k^{d-d_1}/\delta^d)$, matching Haussler's bound for the special cases where $d_1=d$ or $k=n$. As an immediate consequence we conclude that set systems of halfspaces, balls, and parallel slabs defined over $n$ points in $d$-space admit better packing numbers when $k$ is smaller than $n$. Last but not least, we describe applications to (i) spanning trees of low total crossing number, and (ii) geometric discrepancy, based on previous work by the author.