Shallow Packings in Geometry
Esther Ezra

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 be a finite set system defined over an -point set ; we view as a set of indicator vectors over the -dimensional unit cube. A -separated set of is a subcollection , s.t. the Hamming distance between each pair is greater than , where is an integer parameter. The -packing number is then defined as the cardinality of the largest -separated subcollection of . Haussler showed an asymptotically tight bound of on the -packing number if has VC-dimension (or \emph{primal shatter dimension}) . We refine this bound for the scenario where, for any subset, of size and for any parameter , the number of…
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Digital Image Processing Techniques · Topological and Geometric Data Analysis
