{"paper":{"title":"Solution of the propeller conjecture in $\\mathbb{R}^3$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.MG"],"primary_cat":"cs.CC","authors_text":"Assaf Naor, Aukosh Jagannath, Steven Heilman","submitted_at":"2011-12-13T18:27:03Z","abstract_excerpt":"It is shown that every measurable partition ${A_1,..., A_k}$ of $\\mathbb{R}^3$ satisfies $$\\sum_{i=1}^k||\\int_{A_i} xe^{-\\frac12||x||_2^2}dx||_2^2\\le 9\\pi^2.\\qquad(*)$$ Let ${P_1,P_2,P_3}$ be the partition of $\\mathbb{R}^2$ into $120^\\circ$ sectors centered at the origin. The bound is sharp, with equality holding if $A_i=P_i\\times \\mathbb{R}$ for $i\\in {1,2,3}$ and $A_i=\\emptyset$ for $i\\in \\{4,...,k\\}$ (up to measure zero corrections, orthogonal transformations and renumbering of the sets $\\{A_1,...,A_k\\}$). This settles positively the 3-dimensional Propeller Conjecture of Khot and Naor (FOCS"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.2993","kind":"arxiv","version":2},"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"}