{"paper":{"title":"Krivine schemes are optimal","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Assaf Naor, Oded Regev","submitted_at":"2012-05-29T16:38:03Z","abstract_excerpt":"It is shown that for every $k\\in \\N$ there exists a Borel probability measure $\\mu$ on $\\{-1,1\\}^{\\R^{k}}\\times \\{-1,1\\}^{\\R^{k}}$ such that for every $m,n\\in \\N$ and $x_1,..., x_m,y_1,...,y_n\\in S^{m+n-1}$ there exist $x_1',...,x_m',y_1',...,y_n'\\in S^{m+n-1}$ such that if $G:\\R^{m+n}\\to \\R^k$ is a random $k\\times (m+n)$ matrix whose entries are i.i.d. standard Gaussian random variables then for all $(i,j)\\in {1,...,m}\\times {1,...,n}$ we have\n\\E_G[\\int_{{-1,1}^{\\R^{k}}\\times {-1,1}^{\\R^{k}}}f(Gx_i')g(Gy_j')d\\mu(f,g)]=\\frac{<x_i,y_j>}{(1+C/k)K_G},\nwhere $K_G$ is the real Grothendieck constant"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.6415","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"}