{"paper":{"title":"Shallow Packings in Geometry","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"cs.CG","authors_text":"Esther Ezra","submitted_at":"2014-12-16T22:19:40Z","abstract_excerpt":"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 asy"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.5215","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}