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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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1703.09973","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-03-29T11:21:34Z","cross_cats_sorted":[],"title_canon_sha256":"594bc021a9ba9dd65d16dc1ae257f5808369c094b3915c196ddb59ac67ca68a6","abstract_canon_sha256":"4b31233166d98d420ef2c3ee6381ef70a3807a29fa1d461db12081daf7ff99dd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:47:38.907300Z","signature_b64":"BDhAadsOKzWZvC0ChVUyo5ntdT1zVZEMhOsqePnDQ+0IPjpEB6Om1cBqWgoj+lkQr3BnymHOF1trxXdzMoSjAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"659394906020ee55c9679f0feec07edc372df954119e598fabbe7aec4e5a55e7","last_reissued_at":"2026-05-18T00:47:38.906749Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:47:38.906749Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $1\\leq k\\leq\\sqrt n$. 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