{"paper":{"title":"Optimal Vector Balancing for Zonotopes","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.MG","authors_text":"Victor Reis","submitted_at":"2026-05-22T17:25:00Z","abstract_excerpt":"A zonotope is a linear image of the cube $[-1,1]^m$ for some $m \\in \\mathbb{N}$. We show that there is a universal constant $C$ such that, for every zonotope $Z\\subset \\mathbb{R}^d$ and vectors $v_1,\\dots,v_n\\in Z$, there are signs $x_1,\\dots,x_n\\in\\{-1,1\\}$ with \\[\n  \\sum_{i=1}^n x_i v_i \\in C\\sqrt d\\, Z. \\] This resolves a 2002 question of Schechtman and generalizes Spencer's six standard deviations theorem, which corresponds to the case $Z=[-1,1]^d$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.23866","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.23866/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}