{"paper":{"title":"The variance conjecture on projections of the cube","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"David Alonso-Guti\\'errez, Julio Bernu\\'es","submitted_at":"2017-03-29T11:21:34Z","abstract_excerpt":"We prove that the uniform probability measure $\\mu$ on every $(n-k)$-dimensional projection of the $n$-dimensional unit cube verifies the variance conjecture with an absolute constant $C$ $$\\textrm{Var}_\\mu|x|^2\\leq C \\sup_{\\theta\\in S^{n-1}}{\\mathbb E}_\\mu\\langle x,\\theta\\rangle^2{\\mathbb E}_\\mu|x|^2, $$ provided that $1\\leq k\\leq\\sqrt n$. We also prove that if $1\\leq k\\leq n^{\\frac{2}{3}}(\\log n)^{-\\frac{1}{3}}$, the conjecture is true for the family of uniform probabilities on its projections on random $(n-k)$-dimensional subspaces."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.09973","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"}